INTEGRABILITY VIA GEOMETRY: DISPERSIONLESS DIFFERENTIAL EQUATIONS IN THREE AND FOUR DIMENSIONS

DAVID M.J. CALDERBANK AND BORIS KRUGLIKOV

Abstract. We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein– Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.

Introduction and main results

The integrability of dispersionless partial differential equations is well known to admit a geometric interpretation. Twistor theory [30, 27] gives a framework to visualize this for several types of integrable systems, as demonstrated by many examples [31, 20, 15, 38, 2].

Recently, such a relation has been established for several classes of second order equations in 3D and one class in 4D [18]. Namely the following equivalences have been established:

Integrability via hydrodynamic reductions

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Dispersionless Lax pair with spectral parameter

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Integrable background geometry

Hydrodynamic integrability in 2D (also written “1 + 1 dimension”) was introduced in [10] and elaborated in [36]. Integrability via hydrodynamic reductions in d ⩾ 3 dimensions was developed in [17]. This method, although constructive, is not universal, as it applies only to translation invariant equations (invariantly, this requires the existence of a ddimensional abelian contact symmetry group). Thus the upper part of the above diagram, at least at present, does not extend to the general class of second order PDEs.

On the other hand, the two other ingredients of the diagram are universal. The main aim of this paper is to prove the bottom equivalence for large class of PDE systems, including general second order PDEs, in 3D and 4D, where “integrable background geometry” means that a canonical conformal structure on solutions of the equation is Einstein–Weyl in 3D and self-dual in 4D (these geometries are “backgrounds” for integrable gauge theories [2]).

Consider a second order PDE

E : F (x, u, ∂u, ∂^{2}u) = 0  (1)

for a scalar function u of an independent variable x on a connected manifold M with dim M = d, where ∂u = (u_{i}) and ∂^{2}u = (u_{ij}) denote partial derivatives of u in local coordinates x = (x^{i}). Let M_{u} denote the manifold M equipped with a given scalar function u; concretely, we may view M_{u} as the graph of u in M × ℝ. A tensor on M_{u} is, by definition, a tensor on M, which may also depend, at each x ∈ M, on finitely many derivatives of u at x.

Key words and phrases. Integrable system, dispersionless Lax pair, characteristic variety, Einstein– Weyl geometry, self-duality, twistor theory.

Let σ_{F} be the linearization of F in second derivatives, i.e.,

σ_{F} = ∑_{i⩽j} \frac{∂F}{∂u_{ij}} ∂_{i}∂_{j} = ∑_{i,j} σ_{ij}(u) ∂_{i} ⊗ ∂_{j}, where σ_{ij}(u) := \frac{1 + δ_{ij}}{2} \frac{∂F}{∂u_{ij}} .

Invariantly, σ_{F} defines a section of S^{2}T M_{u}, hence a quadratic form on T_{x}^{∗}M_{u} for each x ∈ M_{u}, called the symbol of F. If we change the defining function F of E, σ_{F} changes by a conformal rescaling on E. Hence the conformal class of σ_{F} along F = 0 is an invariant of E, as is the characteristic variety χ^{E} → M_{u}, the bundle whose fibre at x ∈ M_{u} is the projective variety χ_{x}^{E} := Char(E, u)_{x} = {[θ] ∈ ℙ(T_{x}^{∗}M_{u}) | σ_{F} (θ) = 0}.

We assume henceforth that (1) is:

• nondegenerate, i.e., σ_{F} is nondegenerate at generic points of the zero-set E of F. This is equivalent to det(σ_{ij}(u)) ≠ 0 for a generic solution u.

• hyperbolic, i.e., M is complex and F is holomorphic, or M is real, F is smooth and the variety {[θ] ∈ ℙ(T^{∗}M_{u} ⊗ ℂ) | σ_{F} (θ) = 0} of complex characteristics is a complexification of χ^{E} for a generic (real) solution u.

The nondegeneracy of σ_{F} implies that its inverse

g_{F} = ∑_{i,j} g_{ij}(u) dx^{i} dx^{j}, where (g_{ij}(u)) = (σ_{ij}(u))^{−1},

defines a nondegenerate symmetric bilinear form on T_{x}M_{u} for any (x, u) sufficiently close to a generic point of F = 0. As in [18], the corresponding conformal structure c_{F} plays a central role in this paper. Hyperbolicity implies that along F = 0, c_{F} is uniquely determined by the bundle χ^{E} of nonsingular quadric hypersurfaces because the latter is dual to the projectivized null cone of c_{F}.

A dispersionless Lax pair [39] or dLp for (1) can be described as rank one covering system [37] of E. Roughly speaking, this means that there is a fibre bundle π̂: M̂_{u} → M_{u} with connected rank one fibres, and a PDE system on M̂_{u} with E as a differential corollary. There are various ways to formulate this precisely; in this paper we adopt as a definition that there are linearly independent vector fields X̂ and Ŷ on M̂_{u}, whose coefficients depend on finitely many derivatives of u, such that E is the Frobenius integrability condition for their span Π̂ ⊆ T M̂_{u}—this is the condition that [X̂, Ŷ] is a section of Π̂, so that Π̂ is tangent to a foliation of M̂_{u} by surfaces.

The leaf space of this foliation (for a solution u of E) is sometimes called the twistor space Tw of the dLp in 4D (or minitwistor space in 3D). However, a well-behaved twistor space may only exist over suitable open subsets of M_{u}, so its geometry is more conveniently described on the correspondence space M̂_{u}. For instance, functions on Tw correspond to solutions of a linear PDE system for functions on M̂_{u} that are constant on the leaves of the foliation, while hypersurfaces in Tw may be described as solutions of a quasilinear PDE system for sections of π̂: M̂_{u} → M_{u} that are unions of such leaves. Either of these PDE systems can equivalently be called a dLp: E ensures their compatibility.

A fibre coordinate λ: M̂_{u} → ℝ is called a spectral parameter and it locally identifies M̂_{u} with M_{u} × ℝ. We may then write X̂ = X + m ∂_{λ}, Ŷ = Y + n ∂_{λ} where X, Y are λ-parametric vector fields on M_{u}, and a section of π̂: M̂_{u} → M_{u} may be written λ = q(x) for a function q: M_{u} → ℝ. The dLp Π̂ then has the geometric interpretation that E is the integrability condition for the existence of many foliations of M_{u} by surfaces which are tangent at any x ∈ M_{u} to the span Π = π̂_{∗}(Π̂) of X and Y at x, with λ = q(x).

A fundamental motivation for this paper is that in all known examples of such dLps, it has been observed (see e.g. [18]) that Π is characteristic for E in the sense that for any 1-form θ on M_{u} with Π ⊆ ker θ, we have [θ] ∈ χ^{E}. Thus for any solution u of E, M_{u} admits many foliations by characteristic surfaces, and indeed E is the integrability condition for their existence. Our first result establishes this characteristic property in considerable generality.

Theorem 1. Let Π̂ be a dLp on π̂: M̂_{u} → M_{u} for a determined PDE system E of order ℓ on M_{u}. Then Π = π̂_{∗}(Π̂) is characteristic for E.

We refer to Sections 1 and 2, or [22, 23, 37], for discussion of more general PDE systems and their characteristic varieties: in this introduction, we focus on second order scalar PDEs. For such PDEs, the characteristic condition means that for each solution u and each x̂ ∈ M̂_{u}, Πx̂ is a coisotropic 2-plane for the conformal structure c_{F}. By nondegeneracy of c_{F}, such 2-planes can only exist for 2 ⩽ d ⩽ 4: for d = 2, the condition is vacuous; for d = 3, Πx̂ is then tangent to the null cone of c_{F} (i.e., degenerate); for d = 4, Πx̂ is then contained in the null cone (i.e., totally isotropic). In the real case, the characteristic condition further implies that c_{F} has (up to sign) signature (2, 1) for d = 3 or (2, 2) for d = 4. We assume this henceforth.

For both d = 3 and d = 4, the coisotropic 2-planes at each point x ∈ M form a 1-dimensional submanifold of the grassmannian Gr_{2}(T_{x}M). For d = 3 this submanifold is a rational curve (≅ ℙ^{1}, the projective line) canonically isomorphic to the conic χ^{E} ⊆ ℙ(T_{x}^{∗}M). For d = 4, it is a disjoint union of two rational curves, corresponding to the two rulings of the quadric surface χ^{E}; the points of the two components are called α-planes and β-planes depending on whether the 2-planes are self-dual or anti-self-dual.

If Π is coisotropic and is also an immersion, we may thus identify M̂_{u} locally with the ℙ^{1}-bundle whose fibre over x ∈ M_{u} consists of all coisotropic 2-planes for d = 3 or the αplane component for d = 4. Under this identification, Π → M̂_{u} becomes the tautological bundle of coisotropic 2-planes. Any Weyl connection ∇ on M_{u} (a torsion-free conformal connection on M depending on finitely many derivatives of u) induces a connection on M̂_{u} → M_{u} and hence a horizontal lift of Π to distribution Π̂_{∇} ⊆ T M̂_{u}.

If d = 4, it is well-known [30] that Π̂_{∇} is independent of ∇ (i.e., conformally invariant), and is integrable if and only if (M_{u}, c_{F}) is is self-dual (SD), i.e., the Weyl tensor W_{c_{F}} satisfies W_{c_{F}} = ∗W_{c_{F}}. The integral surfaces of Π̂_{∇} then project to α-surfaces for c_{F}.

If d = 3, it is similarly well known [6, 20] that Π̂_{∇} is integrable if and only if (M_{u}, c_{F} , ∇) is Einstein–Weyl (EW), i.e., the symmetrized Ricci tensor of ∇ is proportional to any metric g_{F} in the conformal class: Sym(Ric^{∇}) = Λ g_{F}, Λ ∈ C^{∞}(M_{u}). The integral surfaces of Π̂_{∇} then project to totally geodesic null surfaces for (c_{F} , ∇).

A dLp Π̂ for E arising in this way for d = 3, 4 will be called standard. Two dispersionless Lax pairs Π̂, Π̂′ will be called E-equivalent, if Π̂ = Π̂′ on M̂_{u} for any solution u of E.

It is an open question in the theory of integrable systems how many non-equivalent coverings a given E can possess. Our second result claims that coverings of dLp type are essentially unique under a certain nondegeneracy condition on Π̂. This condition, given in Definition 7 of Section 3.5, depends only on Π = π̂_{∗}(Π), implies that Π immerses, and holds in all examples we know of.

The result is straightforward when d = 4, but when d = 3, it shows that Π̂ can be assumed projective: for some choice of spectral parameter λ and vector fields X̂, Ŷ generating Π̂, the coefficients of these vector fields are cubic polynomials in λ. The result is again not restricted to second order scalar PDEs: we require only that χ_{x}^{E} is a nonsingular quadric hypersurface for each x ∈ M_{u}.

Theorem 2. Let E : F = 0 be a determined PDE system of order ℓ whose characteristic variety χ^{E} is a bundle of nonsingular quadric hypersurfaces in ℙ(T^{∗}M_{u}). Then any nondegenerate dLp Π̂ is E-equivalent to a standard dLp Π̂_{∇} for some Weyl connection ∇.

Our third (and main) result establishes an equivalence between the dispersionless integrability of E and the EW/SD property of c_{F}. However, to achieve this, some care is needed in the formulation of both properties. First, in the integrability of the dLp Π̂, we must account for E-equivalence. Thus we say that E is integrable by a dLp Π̂ if for any Π̂′, which is E-equivalent to Π̂, the Frobenius integrability condition for Π̂′ is a nontrivial differential corollary of E. Secondly, the EW/SD property should be a nontrivial differential corollary of E. The need for nontriviality here is illustrated by PDEs of the form Δu = f(x, u, ∂u): this is non-integrable for generic f, but its conformal structure is independent of u and is flat, so the EW/SD property holds automatically. For more general PDE systems E, a differential corollary of E holds nontrivially if it is not a consequence of a proper subsystem E′ of E. We can now obtain the main result as follows.

Theorem 3. Let E : F = 0 be a determined PDE system in 3D or 4D whose characteristic variety χ^{E} is a bundle of nonsingular quadric hypersurfaces, for instance a nondegenerate hyperbolic second order scalar PDE (1). Let c_{F} be the corresponding conformal structure. Then E is integrable by a nondegenerate dLp if and only if

3D: the Einstein–Weyl property for c_{F} holds nontrivially on solutions of E; 4D: the self-duality property for c_{F} holds nontrivially on solutions of E.

Proof. As a preliminary, note that if F has order ℓ, then c_{F} depends pointwise only on derivatives of u of order ⩽ ℓ (or ⩽ (ℓ − 1) if F is quasilinear) and so is defined and is nondegenerate for almost any u (not necessarily a solution). Thus Π̂_{∇} is defined for any Weyl connection ∇ over an open subset of M_{u}, and its integrability there is equivalent to the EW condition for (c_{F} , ∇) when d = 3 and the SD condition for c_{F} when d = 4.

Suppose first that Π̂ ⊆ T M̂_{u} is a dLp for E. By Theorem 1, Π = π̂_{∗}(Π̂) is characteristic, i.e., when F = 0, Π is coisotropic for the conformal structure c_{F} (and for d = 4 we orient M_{u} so that Π is a congruence of α-planes). Nondegeneracy of Π̂ implies that Π immerses into Gr_{2}(T M_{u}) and so we may assume that M̂_{u} is an open subset of the bundle of coisotropic 2-planes for all solutions u, and hence also on an open neighbourhood of (x, u) where c_{F} is nondegenerate. Then by Theorem 2, Π̂ is E-equivalent to a standard dLp Π̂_{∇} over any open subset of M_{u}. Hence the EW/SD condition is a nontrivial differential corollary of E, as required.

Conversely, suppose that the EW/SD condition is a nontrivial differential corollary of E (for some Weyl connection ∇ when d = 3), and let π̂: M̂_{u} → M_{u} be the bundle of null 2-planes for d = 3, or the bundle of α-planes for d = 4. Then if Π̂ is E-equivalent to Π̂_{∇} (for any Weyl connection ∇ when d = 4) on an open subset of M_{u}, the integrability of Π̂ is a differential corollary of E on that open subset (since this is true for Π̂_{∇}).

Finally if any such Π̂ is a differential corollary of a proper subsystem E′ of E, then the first part of the argument implies that the EW/SD property is also a consequence of E′, contradicting nontriviality. □

Remark 1. Often, in the physics literature, little distinction is made between a system E and a system E′ obtained by differentiation or potentiation of E. While some properties of the equation can change, for instance the symmetry algebra and dimension of the solution space, the characteristic variety and integrability of E are unaltered. It is easy to adjust the formulation of the theorems to such variations between E and E′.

This theorem shows that the EW and SD equations are master equations, in 3D and 4D respectively, for determined integrable PDE systems whose characteristic variety is a bundle of nonsingular quadric hypersurfaces. It applies in particular to first order systems and higher order scalar equations whose (principal) symbol is a power of a nondegenerate quadratic form. However, the EW and SD equations are not themselves determined systems because of the gauge freedom coming from diffeomorphism invariance. Determined forms of the EW and SD equations were derived in [13], where it was shown in particular that the Manakov–Santini system [25] is equivalent to a determined form of the EW equation. Because of their importance, we will present novel derivations of these determined master equations using the methods of this paper.

Theorem 3 is useful for at least two reasons. First, the geometric characterizations of integrability are algorithmic. In 4D, the anti-self-dual part of the Weyl tensor of c_{F} on M_{u} can be computed explicitly from finitely many derivatives of u, and so we can check whether it vanishes on solutions by imposing the equation and its prolongations formally—we do not have to be able to resolve the PDE or even to prove its solvability. In 3D, the situation is complicated slightly by the choice of Weyl connection. For the classes of translation-invariant equations considered in [18], there is a universal formula for the Weyl connection, but this formula is not generally applicable (it is not contact-invariant). Nevertheless, except in degenerate situations, the choice is uniquely determined by finitely many derivatives of c_{F}, and so the EW condition may again be verified by formally imposing the PDE on a tensor depending on finitely many derivatives of u. This effective integrability criterion has many applications: for instance, it was applied in [24] to obtain infinitely many new integrable equations in 4D as deformations of integrable Monge-Ampère equations of Hirota type.

Secondly, the EW/SD property provides a canonical characteristic Lax pair, which, if the PDE on u has order ℓ, depends on at most ℓ + 1 derivatives of u (ℓ if the PDE is quasilinear), and satisfies a ‘normality’ condition off shell which is useful in computations. None of these properties were assumed a priori. For example, the standard Lax pair [25] for the Manakov–Santini system is not normal, and the normal Lax pair may be understood as a Lax pair for an equivalent PDE system presented in [13], which we also discuss.

Apart from the Manakov–Santini system (and variants), Theorem 3 encompasses many examples in 3D, such as the Lax pairs arising in the central quadric ansatz [16], for EW manifolds in diagonal coordinates [13], and for the systems of two first order PDE on two unknown functions studied in [8]. In 4D, there are Lax pairs having no derivatives with respect to the spectral parameter λ, which cannot be normal, such as the hypercomplex Lax pair of Dunajski and Joyce (see [2, 13]) and Lax pairs for Monge-Ampère equations of Hirota type [7]. However, normal Lax pairs are always available, and provide a canonical choice in 4D, while in 3D they are given by a choice of Weyl connection.

We begin the body of the paper in Section 1 by presenting a rigorous definition of what should be called a (nondegenerate) dispersionless Lax pair, motivated by examples. The search for such formalism in general has a long history: see [4, 26] for discussion in the dispersive context. A fundamental role is played by the λ-dependent family Π = π̂_{∗}(Π̂) of rank 2 subbundles of T M_{u}, which we call a 2-plane congruence. We also explain the normality condition mentioned above, observing that in 4D it determines Π̂ from Π.

In Section 2, we prove Theorem 1. Here we treat the symbol and characteristic variety of general PDE systems. For both this, and the proof of Theorem 2, we require some jet theory, which we have generally suppressed in the rest of the paper, cf. Remark 2. Having proven Theorem 1, as an addendum, we show in Section 2.3 that a Lax pair which is characteristic for a quadric is nondegenerate, and give a computational criterion for the existence of such a quadric for nondegenerate Lax pairs.

For PDE systems whose characteristic variety is a quadric, Theorem 1 shows that Π is essentially unique, which considerably constrains the choice of Π̂, especially in 4D. In 3D, however, more work is needed to prove Theorem 2, which we develop in Section 3. We first discuss the standard EW/SD Lax pairs, which are not only normal, but projective. We also introduce and motivate a stronger nondegeneracy condition on the Lax pair Π̂. Roughly speaking, this condition means that the equation E appears nontrivially in the symbol of the integrability condition for Π̂ (i.e., at highest order). From this we deduce the projective property of the Lax pair, and hence prove Theorem 2.

In Section 4 we discuss applications and extensions of the viewpoint we have developed. In particular, we discuss pseudopotentials and their relation to contact coverings, the twistor interpretation of this relationship, and potential generalizations of the theory.

1. Lax pairs: nondegeneracy and normalization

1.1. Dispersionless pairs and 2-plane congruences. We begin with a well-known prototypical example.

Example 1 (dKP). The dispersionless Kadomtsev–Petviashvilli (dKP) equation (see for example [15]) is the second order scalar PDE

F (x, u, ∂u, ∂^{2}u) := u_{xt} + (uu_{t})_{t} − u_{yy} = 0  (2)

for a scalar function u on a 3-manifold M_{u} ≃ M with coordinates (x, y, t). (This differs from some standard conventions by the interchange t ↔ x and/or u ↦ −u.) The dKP equation is the compatibility condition ψ_{xy} = ψ_{yx} of the first order linear system

ψ_{x} − (λ^{2} − u) ψ_{t} − (u_{y} + λu_{t}) ψ_{λ} = 0, ψ_{y} − λ ψ_{t} − u_{t} ψ_{λ} = 0,

for a scalar function ψ on M̂_{u} = M_{u} × ℝ with coordinates (x, y, t, λ). It may also be described as the compatibility condition q_{xy} = q_{yx} for the quasilinear system

q_{x} = (q^{2} − u) q_{t} − q u_{t} − u_{y}, q_{y} = q q_{t} − u_{t}

for a scalar function q = q(x, y, t) on M_{u}. In more geometric terms, ψ is a function on M̂_{u} which is invariant under the vector fields

X̂ = ∂_{x} − (λ^{2} − u) ∂_{t} − (u_{y} + λu_{t}) ∂_{λ},  Ŷ = ∂_{y} − λ ∂_{t} − u_{t} ∂_{λ},  (3)

while q defines a section of π̂: M̂_{u} → M_{u} such that X̂ and Ŷ are tangent to its image. The compatibility condition in either case is that X̂ and Ŷ span a distribution Π̂ ⊆ T M̂_{u} which is (Frobenius) integrable, i.e., [X̂, Ŷ] is also section of Π̂. In this example, the Frobenius integrability condition holds if and only if [X̂, Ŷ] = 0 if and only if (2) is satisfied.

In this paper, we take the distribution Π̂ on M̂_{u} to be the fundamental object.

Definition 1. A dispersionless pair of order ⩽ N is a bundle π̂: M̂_{u} → M_{u} called the correspondence space, whose fibres are connected curves, together with a rank two distribution Π̂ ⊆ T M̂_{u} such that:

• for all x̂ ∈ M̂_{u}, Π̂x̂ ⊆ Tx̂M̂_{u} depends on u only through its partial derivatives at x = π̂(x̂) ∈ M_{u} of order ⩽ N;

• Π̂ is transverse to the fibres of π̂, i.e., Π̂ ∩ ker π̂_{∗} = 0. A spectral parameter is a local fibre coordinate λ = λ(x̂): M̂_{u} → ℝ on M̂_{u}.

If Π̂ = ⟨X̂, Ŷ ⟩, we thus obtain a first order linear system

X̂(ψ) = 0,  Ŷ (ψ) = 0  (4)

for functions ψ on M̂_{u}. In terms of a spectral parameter λ, a section of π̂ has image λ = q(x) for a function q: M_{u} → ℝ, and the corresponding first order quasilinear system is given by

X̂(λ − q(x))|_{λ=q(x)} = 0, Ŷ (λ − q(x))|_{λ=q(x)} = 0. (5)

The system (4) is compatible if and only if (5) is compatible if and only if the distribution Π̂ is integrable. Then solutions of (4) and (5) describe respectively functions and hypersurfaces in the (local) leaf space of the folation tangent to Π̂ (the twistor or minitwistor space). The integrability condition of Π̂ is a PDE on u of order ⩽ N + 1. Roughly speaking—see Definition 5—dispersionless integrable systems are PDEs arising as such integrability conditions.

We need not restrict attention to scalar PDEs. Indeed we wish to encompass the following important system due to Manakov and Santini [25].

Example 2 (MS). The Manakov–Santini (MS) system is the second order coupled system of PDEs

S(u) + u_{t}^{2} = 0, S(v) = 0 (6)

for functions (u, v) of (x, y, t), where

S = ∂_{t}∂_{x} + v_{t} ∂_{t}∂_{y} + (u − v_{y}) ∂_{t}^{2} − ∂_{y}^{2}.  (7)

(As with the dKP equation, we have aligned our coordinate conventions for consistency within this paper. Conventions in the literature [25, 13, 29] vary, but are all equivalent to the one here by point transformations.)

As noted in [25], system (6) is the Frobenius integrability condition for the dispersionless pair Π̂ = ⟨X̂, Ŷ ⟩ spanned by

X̂ = ∂_{x} − (λ^{2} + v_{t}λ − u + v_{y}) ∂_{t} − (u_{t}λ + u_{y}) ∂_{λ},  Ŷ = ∂_{y} − (λ + v_{t}) ∂_{t} − u_{t} ∂_{λ}. (8)

The corresponding quasilinear covering system, which was studied in [29] and more recently in [33], is

q_{x} = (q^{2} + q v_{t} − u + v_{y}) q_{t} − q u_{t} − u_{y}, q_{y} = (v_{t} + q) q_{t} − u_{t}.

When v = 0, the MS system reduces to the dKP equation, and (8) to (3). When u = 0, the dLp (8) has no derivatives with respect to the spectral parameter.

If Π̂ is a dispersionless pair, then Π := π̂_{∗}(Π̂) is a rank 2 subbundle of π̂^{∗}T M_{u}, so at each x ∈ M_{u}, we have a 1-parameter family of 2-dimensional subspaces of T_{x}M.

Definition 2. A 2-plane congruence Π over M_{u} is a section Π: M̂_{u} → π̂^{∗} Gr_{2}(T M_{u}), where Gr_{2}(T M_{u}) → M_{u} is the bundle whose fibre over x ∈ M_{u} is the grassmannian of 2-dimensional vector subspaces of T_{x}M_{u}.

Conversely, the passage from a 2-plane congruence Π to a dispersionless pair Π̂ can be understood as a lift with respect to the projection π̂: M̂_{u} → M_{u}. It is convenient to describe the lift condition in terms of the rank 3 distribution Δ = π̂_{∗}^{−1}(Π) ⊆ T M̂_{u}: Π̂ is a lift of Π if and only if it is a rank 2 subbundle of Δ transverse to the fibres of π̂. For any distributions D_{1}, D_{2} ⊆ T M̂_{u} we denote by [D_{1}, D_{2}] the distribution generated by Lie brackets of sections of D_{1} and D_{2}. Thus the integrability condition for Π̂ is that its derived distribution [Π̂, Π̂] is equal to Π̂.

More explicitly, we choose a spectral parameter λ and let X, Y be linearly independent λ-parametric vector fields on M_{u} depending at each x only on the partial derivatives of u at x of order ⩽ N. Then Π = ⟨X, Y ⟩ is a 2-plane congruence, and Δ is the span of the coordinate lifts of X, Y (still denoted X, Y, with X(λ) = 0 = Y (λ)) and ∂_{λ}. Then we write a dispersionless pair Π̂ on M̂_{u}, with π̂_{∗}(Π̂) = Π as the span Π̂ = ⟨X̂, Ŷ ⟩ of vector fields

X̂ = X + m ∂_{λ},  Ŷ = Y + n ∂_{λ}  (9)

with π̂_{∗}(X̂) = X and π̂_{∗}(Ŷ) = Y, where m, n are functions of x, u, and the spectral parameter λ. The derived distribution of Π̂ is now [Π̂, Π̂] = ⟨X̂, Ŷ , [X̂, Ŷ]⟩ ⊆ T M̂_{u}, which generically has rank 3, and the integrability condition is that it has rank 2.

In 3D, we may introduce coordinates (x, y, t) and choose generators of Π of the form

X = ∂_{x} − α ∂_{t},  Y = ∂_{y} − β ∂_{t},  (10)

where the functions α, β depend on (x, y, t), u and λ. Dually, the annihilator Ann(Π) of Π in π̂^{∗}T^{∗}M_{u} is spanned by the λ-dependent 1-form

θ = dt + α dx + β dy,  (11)

Ann(Δ) is spanned by the pullback of θ to M̂_{u} (which we still denote by θ), while Ann(Π̂) is spanned by θ and the 1-form

η = dλ − m dx − n dy  (12)

on M̂_{u}. Hence Π̂ is the radical of the 2-form θ ∧ η.

In 4D, we similarly may assume that we have coordinates (x, y, z, t) and generators

X = ∂_{x} − α ∂_{z} − β ∂_{t},  Y = ∂_{y} − γ ∂_{z} − δ ∂_{t},  (13)

where α, β, γ, δ depend on (x, y, z, t), u and λ. Thus Ann(Π) is spanned by

ζ = dz + α dx + γ dy,  θ = dt + β dx + δ dy,  (14)

Ann(Δ) by their pullbacks, and Ann(Π̂) = ⟨ζ, θ, η⟩ with η given by (12). In both 3D and 4D, with X̂ and Ŷ given by (9), Π̂ is integrable if and only if [X̂, Ŷ] = 0.

1.2. Normality and nondegeneracy. In order for Π̂ to be a dispersionless Lax pair for an equation E : F = 0, we require that the integrability condition [Π̂, Π̂] = Π̂ holds modulo E, i.e., when F = 0 or, to use physics terminology, on shell.

Definition 3. We say that the dispersionless pair Π̂ ⊆ T M̂_{u} is normal if [Π̂, Π̂] ⊆ Δ off shell, i.e., without assuming F = 0. In other words, π̂_{∗}([Π̂, Π̂]) = Π.

If Π̂ = ⟨X̂, Ŷ ⟩ with X̂ and Ŷ defined by (9), (10) and (13), then Π̂ is normal if and only if [X̂, Ŷ] is a multiple of ∂_{λ}. In this case the integrability condition reduces to the vanishing of the ∂_{λ}-component X̂(n) − Ŷ (m) of the vector field [X̂, Ŷ] (identically in λ).

When d = 4, a generic 2-plane congruence Π has a unique normal lift. Indeed, generically, Δ is nonholonomic with [Δ, Δ] = T M̂_{u}, i.e., it has the growth vector (3, 5), and following Cartan [5, §11], there is a unique rank 2 subbundle Π̂ ⊆ Δ with [Π̂, Π̂] = Δ. Such rank 2 distribution Π̂ either has the growth vector (2, 3, 5) or is integrable. The former case corresponds to Cartan’s celebrated Pfaffian system [5] (for nonintegrable systems or off shell), the latter case corresponds to a dispersionless Lax pair (on shell).

The genericity condition we need here is as follows (and we formulate a similar condition for d = 3 which we will use later).

Definition 4. A 2-plane congruence Π is called nondegenerate if

θ ∧ θ_{λ} ∧ θ_{λλ} ≠ 0, where Ann(Π) = ⟨θ⟩ for d = 3;
θ ∧ ζ ∧ θ_{λ} ∧ ζ_{λ} ≠ 0, where Ann(Π) = ⟨θ, ζ⟩ for d = 4.

(15)

These conditions depend only on Π, not on the choices of θ or ζ: when d = 4 nondegeneracy means equivalently ϖ_{λ}∧ϖ_{λ} ≠ 0 where ϖ = θ∧ζ, or dually that X ∧Y ∧X_{λ}∧Y_{λ} ≠ 0, where Π = ⟨X, Y ⟩. If we choose θ and ζ as in (11) and (14), then the nondegeneracy conditions may be written explicitly as:

α_{λ}β_{λλ} − α_{λλ}β_{λ} ≠ 0  for d = 3;  (16)
α_{λ}δ_{λ} − β_{λ}γ_{λ} ≠ 0  for d = 4.  (17)

Lemma 4. For d = 4, any nondegenerate 2-plane congruence Π has a unique normal lift.

Proof. If X̂ and Ŷ are given by (9) and (13), dx([X̂, Ŷ]) = 0 = dy([X̂, Ŷ]) identically, while dz([X̂, Ŷ]) = dt([X̂, Ŷ]) = 0 form two linear equations on m, n:

[\begin{matrix}δ_{λ} & −β_{λ} \\ −γ_{λ} & α_{λ} \end{matrix}] [\begin{matrix}m \\ n \end{matrix}] = [\begin{matrix}αδ_{z} + βδ_{t} − γβ_{z} − δβ_{t} + β_{y} − δ_{x} \\ γα_{z} + δα_{t} − αγ_{z} − βγ_{t} + γ_{x} − α_{y} \end{matrix}];

these have a unique solution by the nondegeneracy condition (17). □

Example 3 (SDM). We illustrate this with the master equation for SD structures obtained in [13, Theorem 2]. Consider a 2-plane congruence Π spanned by (13) with α_{λ} = 0 = δ_{λ} and β_{λ} = 1 = −γ_{λ}. This is totally isotropic for the conformal class of the metric

g = θ_{λ} ζ − ζ_{λ} θ = dx (dz + α dx + γ dy) + dy (dt + β dx + δ dy),

which is independent of λ. In particular, there is a foliation by the totally isotropic level surfaces of (x, y). Any SD metric can be written in this form, with the isotropic surface foliation being anti-self-dual [13, 32]. The unique normal lift of Π is given by (9) with

m = γ_{x} − α_{y} + δα_{t} − αγ_{z} + γα_{z} − βγ_{t}, n = δ_{x} − β_{y} + δβ_{t} − αδ_{z} + γβ_{z} − βδ_{t}.

Now the λ^{2} term of the integrability condition X̂(n)−Ŷ (m) = 0 is (α_{z}+γ_{t})_{z}+(β_{z}+δ_{t})_{t} = 0, so α_{z} +γ_{t} = s_{t} and β_{z} +δ_{t} = −s_{z} for some function s. However, we may use the translation freedom in λ to set s = 0, so that α = u_{t}, γ = −(λ + u_{z}), β = λ − v_{t}, δ = v_{z} for functions (u, v) of (x, y, z, t). Thus we obtain a normal dispersionless pair Π̂ = ⟨X̂, Ŷ ⟩ with

X̂ = ∂_{x} − u_{t} ∂_{z} − (λ − v_{t}) ∂_{t} − Q(u)∂_{λ},  Ŷ = ∂_{y} + (λ + u_{z}) ∂_{z} − v_{z} ∂_{t} + Q(v)∂_{λ},
where Q = ∂_{x}∂_{z} + ∂_{y}∂_{t} − u_{t}∂_{z}^{2} + (u_{z} + v_{t})∂_{z}∂_{t} − v_{z}∂_{t}^{2}.

The corresponding quasilinear system (5) is

q_{x} − u_{t}q_{z} − (q − v_{t})q_{t} = −Q(u),  q_{y} + (q + u_{z})q_{z} − v_{z}q_{t} = Q(v),  (18)

and the integrability condition reduces to X(Q(v)) + Y (Q(u)) = 0, i.e.,

∂_{z}(Q(u)) = ∂_{t}(Q(v)),  (∂_{x} − u_{t}∂_{z} + v_{t}∂_{t})Q(v) + (∂_{y} + u_{z}∂_{z} − v_{z}∂_{t})Q(u) = 0. (19)

Up to some minor coordinate changes, this is the SD master equation (SDM) of [13].

1.3. Integrability, dispersionless Lax pairs and normalization. When d = 3, we do not obtain a unique normal lift.

Example 4 (MS). The dispersionless pair (8) for the Manakov–Santini system (6) satisfies

[X̂, Ŷ] = −G ∂_{t} − F ∂_{λ}

with F = S(u) + u_{t}^{2}, G = S(v), and so is not normal. However, if we set X̂′ = X̂ − G ∂_{λ} then X̂′ = X̂ on shell (when F = G = 0), while

[X̂′, Ŷ] = [X̂, Ŷ] − G [∂_{λ}, Ŷ] + Y (G) ∂_{λ} = −(F − G_{y} + (λ + v_{t})G_{t}) ∂_{λ}

so Π̂′ := ⟨X̂′, Ŷ ⟩ is normal, and is integrable if and only if

G_{t} = 0, G_{y} = F,

i.e., G = ψ(x, y) and F = ψ_{y}. However, this system is not substantively different from the Manakov–Santini system itself, because we can make a point transformation u ↦ u − ϕ_{y}(x, y), v ↦ v − ϕ(x, y) and if ϕ_{yy} = ψ, we obtain F = 0, G = 0.

This example illustrates two important issues that we want to incorporate into the definition of a dispersionless Lax pair Π̂ for an equation E: first Π̂ is only determined modulo E, and secondly it can be too restrictive in examples to require that the integrability conditions for a dispersionless pair are equivalent to E.

Definition 5. Let E : F = 0 be a PDE system on u and Π̂ ⊆ T M̂_{u} a dispersionless pair.

• A dispersionless pair Π̂′ ⊆ T M̂_{u} is E-equivalent to Π̂ if Π̂ = Π̂′ whenever F (u) = 0.

• Π̂ is a dispersionless Lax pair (dLp) for E if for any Π̂′ E-equivalent to Π̂, the integrability condition [Π̂′, Π̂′] = Π̂′ is a nontrivial differential corollary of E.

To make precise the notion of a differential corollary, we introduce some jet formalism, for which we refer to [22, 23, 37] for further details. A scalar PDE of order ℓ on a manifold M may be defined as an equation of the form

F (j^{ℓ}u) = 0  (20)

where F ∈ C^{∞}(J^{ℓ}M) is a function on the bundle π_{ℓ}: J^{ℓ}M → M of ℓ-jets of functions u on M, and j^{ℓ}u: M → J^{ℓ}M is the ℓ-jet of u, i.e., in coordinates j^{ℓ}u = (x, u, ∂u, … ∂^{ℓ}u).

In order to discuss objects (such as dLps) depending on an arbitrary finite jet of u, we use the infinite jet bundle π_{∞}: J^{∞}M → M which is the union (inverse limit) of J^{k}M over all k. A function f: J^{∞}M → ℝ is smooth if it is the pullback of a function on J^{k}M for some k ∈ ℕ, in which case we say f has order ⩽ k. A choice of coordinates x^{i} on M leads to coordinates (x^{i}, u_{α}) on J^{∞}M, where 1 ⩽ i ⩽ d and α runs over all symmetric multi-indices in d entries. Then f ∈ C^{∞}(J^{∞}M) has order ⩽ k iff it is a function of x^{i} and u_{α} for all i and α = (i_{1}, … i_{j}) with |α| = j ⩽ k.

The bundle J^{∞}M has a canonical flat connection, the Cartan distribution, for which the horizontal lift of a vector field X on M is the total derivative D_{X} characterized by (D_{X}f) ∘ j^{∞}u = X(f ∘ j^{∞}u) for any smooth function f on J^{∞}M. More generally, any section X of π_{∞}^{∗}T M has a lift to a vector field D_{X} on J^{∞}M, given in local coordinates by D_{X} = ∑_{i} a_{i}D_{i}, where X = ∑_{i} a_{i}∂_{i} and D_{i} = ∂_{i} + ∑_{α} u_{iα}∂_{u_{α}}.

Higher order operators □ in total derivatives (also known as 𝒞 -differential operators) are generated as compositions of the derivations D_{X} with coefficients being smooth functions on J^{∞}M. In local coordinates, □ = ∑ a_{α}D_{α}, where a_{α} ∈ C^{∞}(J^{∞}M) and D_{α} = D_{i_{1}} ⋯ D_{i_{j}} for a multi-index α = (i_{1}, … i_{j}) with entries in {1, 2, … d}.

Let I_{F} be the ideal in C^{∞}(J^{∞}M) generated by the pullback of F ∈ C^{∞}(J^{ℓ}M) and its total derivatives of arbitrary order. Then the zero-set E_{∞} ⊆ J^{∞}M of I_{F} is the space of formal solutions of (20): u is a solution of (20) iff j^{∞}u is a section of E_{∞}.

These notions extend straightforwardly to PDE systems by replacing J^{∞}M with the bundle π_{∞}: J^{∞}(M, V) → M of jets of sections of a fibre bundle V → M, and F by a function of order ⩽ ℓ on J^{∞}(M, V) with values in a vector bundle W → M. The ideal I_{F} in C^{∞}(J^{∞}(M, V)) is now generated by the components of F and their total derivatives of arbitrary order.

In this formalism, a differential corollary of E : F = 0 is a subset of I_{F} (or, more invariantly, the ideal I ⊆ I_{F} generated by this subset and its total derivatives of arbitrary order). It is nontrivial provided it is not a subset of I_{F′} for any F′ whose zero-set in J^{ℓ}(M, V) contains the zero-set of F in positive codimension. For example, the ideal generated by u_{xy}, for a scalar function u(x, y, t), is trivial as a differential corollary of the system F (j^{1}u) := (u_{x}, u_{y}) = 0, because it is a differential corollary of the equation F′(j^{1}u) := u_{x}u_{y} = 0 in which the zero-set of F has positive codimension. However, it is a nontrivial differential corollary of the equation F̃(j^{1}u) := u_{x} = 0.

Consequently, in Definition 5, the integrability condition for a dLp Π̂ for E : F = 0 need not generate I_{F}: indeed, the freedom to replace a dLp by an E-equivalent one may change the ideal I ⊆ I_{F} that its integrability conditions generate.

Remark 2. In most of the paper we make minimal use of the jet formalism by using the philosophy [22, 37] that a differential equation E_{∞} ⊆ J^{∞}M is a generalized manifold whose “points” are solutions u, identified with M_{u} = (j^{∞}u)(M) ⊆ E_{∞} that is diffeomorphic to M via π_{∞}. We are justified in working “pointwise” provided there are enough “points” (i.e., for generic u_{∞} ∈ E_{∞} there is a solution u with u_{∞} ∈ M_{u}), and there are existence theorems for hyperbolic PDEs (or rather, ultrahyperbolic PDEs in signature (2, 2)) which assert this in some generality. Nevertheless, we would rather not rely upon such analytical results here, and all our results can be formalized using jets, even if we do not do so explicitly.

The following normalization result now suffices to establish Theorem 2 when d = 4.

Proposition 5. Let Π̂ be a dLp such that Π = π̂_{∗}(Π̂) is nondegenerate. Then Π̂ is E-equivalent to a normal dLp. Such a Lax pair for d = 4 is unique.

Proof. When d = 4 the Lax pair condition (on shell) implies

dz ∘ π_{∗}[X̂, Ŷ] = □_{1}F, dt ∘ π_{∗}[X̂, Ŷ] = □_{2}F

for some operators □_{1}, □_{2} in total derivatives. Let us modify X̃ = X̂ + A(F)∂_{λ}, Ỹ = Ŷ + B(F)∂_{λ}, where A, B are operators in total derivatives to be determined (they also depend on λ). The new commutation equation is

dz ∘ π_{∗}[X̃, Ỹ] = (□_{1} + α_{λ}B − γ_{λ}A)F
dt ∘ π_{∗}[X̃, Ỹ] = (□_{2} + β_{λ}B − δ_{λ}A)F.

Vanishing of these, equivalent to normality, can be achieved by a unique choice of the operators in total derivatives A, B due to nondegeneracy condition (17).

When d = 3, the Lax pair condition (on shell) implies similarly

dt ∘ π_{∗}[X̂, Ŷ] = Ŷ (α) − X̂(β) = □F

for some operator □ in total derivatives. The modification X̃ = X̂ + A(F)∂_{λ}, Ỹ = Ŷ + B(F)∂_{λ} gives the new commutation relations

dt ∘ π_{∗}[X̃, Ỹ] = (□ + α_{λ}B − β_{λ}A)F.

The equation β_{λ}A − α_{λ}B = □ admits the solution A = α_{λλ}□/(β_{λ}α_{λλ} − α_{λ}β_{λλ}) and B = β_{λλ}□/(β_{λ}α_{λλ}−α_{λ}β_{λλ}) by (16), unique up to the freedom (A, B) ↦ (A, B)+(α_{λ}, β_{λ})L. □

2. The characteristic condition for dispersionless Lax pairs

2.1. Symbols and the characteristic condition. In order to prove Theorem 1 in full generality, we need the notions of symbol and characteristic variety for a general PDE system. For this we use the jet formalism. Recall from the previous section that a smooth function F on J^{∞}M has order ⩽ ℓ if it is a pullback from J^{ℓ}M, and that J^{∞}M has a canonical connection, the Cartan distribution. The vertical part of the 1-form dF ∈ Ω^{1}(J^{∞}M) may be viewed in coordinates as a polynomial on π_{∞}^{∗}T^{∗}M given by

∑_{j=0}^{ℓ} F_{(j)} where F_{(j)} = ∑_{|α|=j}(∂_{u_{α}}F)∂_{α} is a section of π_{∞}^{∗}S^{j}T M.

The top degree term σ_{F} = F_{(ℓ)}, called the (order ℓ) symbol of F, is independent of coordinates. We assume it is nonvanishing: if it vanishes, F has order ⩽ ℓ − 1 and σ_{F} has lower degree.

This generalizes to a PDE system of order ℓ, i.e., a function F of order ⩽ ℓ on J^{∞}(M, V), for some fibre bundle V, with values in a vector bundle W → M. The symbol σ_{F} of F is then a homogeneous degree ℓ polynomial on π_{∞}^{∗}T^{∗}M with values in Hom(T V, W), which we assume is not identically zero, so that the PDE system does not have order ⩽ ℓ − 1. The characteristic variety of the PDE system E : F = 0 is defined by [35]

χ^{E} = {[θ] ∈ ℙ(π_{∞}^{∗}T^{∗}M) | σ_{F} (θ) is not injective}.

If V and W have the same rank, then [θ] is characteristic iff σ_{F} (θ) is not surjective. We take rank(V) = rank(W) as the definition of a determined system, although a more proper definition is codim χ^{E} = 1.

Definition 6. We say that a 2-plane congruence Π (or a dLp Π̂) is characteristic for E if for any solution u of E and any θ in Ann(Π) ⊆ π̂^{∗}T^{∗}M_{u}, we have [θ] ∈ χ^{E}.

In the jet formalism, a dispersionless pair Π̂ lives on a rank 1-bundle π̂: M̂ → J^{∞}(M, V) (so that M̂_{u} = (j^{∞}u)^{∗}M̂) and we let π̂_{∞} = π_{∞} ∘ π̂: M̂ → M. A 2-plane congruence Π is then a rank 2 subbundle of π̂_{∞}^{∗}T M, and Π̂ is a lift of Π to T M̂. In practice we use a spectral parameter λ to trivialize M̂ over J^{∞}(M, V). Then T M̂ is the direct sum of the vertical bundle of π̂, spanned by ∂_{λ}, and π̂^{∗}T J^{∞}(M, V). Thus if Π is spanned by X, Y ∈ π̂_{∞}^{∗}T M, we may write the dispersionless pair Π̂ as the span of X̂ = D_{X} + m ∂_{λ} and Ŷ = D_{Y} + n ∂_{λ}, where D_{X} and D_{Y} are total derivatives (depending also on λ) and m, n are functions on M̂. Then

[X̂, Ŷ] = ([D_{X}, D_{Y}] + m D_{∂_{λ}Y} − n D_{∂_{λ}X}) + (D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m) ∂_{λ}.

The integrability condition [X̂, Ŷ] ∈ Γ(Π̂) reduces to [X, Y] + m ∂_{λ}Y − n ∂_{λ}X = ν_{X} X + ν_{Y} Y, for some ν_{X}, ν_{Y}, together with the vanishing of D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m − ν_{X} m − ν_{Y} n. As in the previous section, we may choose X and Y so that ν_{X} = ν_{Y} = 0, and hence the Lax equation (split into the vertical and horizontal parts) becomes the system

D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m = 0,  (21)
D_{X}Y − D_{Y} X + m ∂_{λ}Y − n ∂_{λ}X = 0.  (22)

We thus have a dLp for E if these equations hold modulo I_{F} i.e., all components (and hence their total derivatives of arbitrary order) belong to I_{F}.

Lemma 6. If D_{X}q − D_{Y} p has order ⩽ k, for functions p, q of u_{∞} ∈ J^{∞}(M, V) and sections X, Y of π_{∞}^{∗}T M, then its order k symbol is

D_{X_{(k)}}q + D_{X}(q_{(k)}) + X ⊙ q_{(k−1)} − D_{Y_{(k)}}p − D_{Y} (p_{(k)}) − Y ⊙ p_{(k−1)}.  (23)

If X, Y are linearly independent, and P_{1} and P_{2} are symmetric k-vectors with X ⊙ P_{2} = Y ⊙ P_{1}, there is a symmetric (k − 1)-vector S with P_{1} = X ⊙ S and P_{2} = Y ⊙ S.

Proof. Equation (23) is straightforward from the definition of the total derivative and the product rule for the vertical differentiation. Extending X, Y pointwise to a basis, the second part reduces to the trivial observation that for any homogeneous polynomials P_{j} = P_{j}(ξ_{1}, … ξ_{d}), j = 1, 2, with ξ_{1}P_{2} = ξ_{2}P_{1}, there is a homogeneous polynomial P with P_{j} = ξ_{j}P. □

Lemma 7. Let (21)–(22) have order ⩽ k + 1 modulo I_{F}, i.e., all their higher symbols vanish modulo I_{F}. Then there is a symmetric k-tensor S_{k} and a symmetric T M-valued k-tensor Q_{k} such that, modulo I_{F}, the order k+1 symbols of (21) and (22) are respectively

\begin{matrix}X ⊙ (n_{(k)} + D_{Q_{k}}n − D_{Y} S_{k} + S_{k} ∂_{λ}n − n ∂_{λ}S_{k}) \\ − Y ⊙ (m_{(k)} + D_{Q_{k}}m − D_{X}S_{k} + S_{k} ∂_{λ}m − m ∂_{λ}S_{k}), \end{matrix} (24)
\begin{matrix}X ⊙ (Y_{(k)} + D_{Q_{k}}Y − D_{Y} Q_{k} + S_{k} ∂_{λ}Y − n ∂_{λ}Q_{k}) \\ − Y ⊙ (X_{(k)} + D_{Q_{k}}X − D_{X}Q_{k} + S_{k} ∂_{λ}X − m ∂_{λ}Q_{k}). \end{matrix} (25)

Proof. Suppose that X, Y, m, n depend only on the N-jet of u for some N ∈ ℕ, so that (21)–(22) have order ⩽ N + 1, and it suffices to prove the lemma for k ⩽ N. We thus induct on p = N − k. For p = 0, the order k + 1 = N + 1 symbols of (21) and (22) are simply X ⊙ n_{(k)} − Y ⊙ m_{(k)} and X ⊙ Y_{(k)} − Y ⊙ X_{(k)} by (23), so we are done, with S_{k} = 0 = Q_{k}.

Now suppose that the lemma holds with k = N − p for some p ⩾ 0, and suppose that (21)–(22) have order ⩽ k modulo I_{F}. Then (21) certainly has order ⩽ k + 1 modulo I_{F}, and so the inductive hypothesis implies its order k +1 symbol, which vanishes modulo I_{F}, is given by (24). Hence Lemma 6 produces a symmetric (k −1)-tensor S_{k−1} such that,

INTEGRABILITY VIA GEOMETRY 13

modulo I_{F},

m_{(k)} = X ⊙ S_{k−1} − D_{Q_{k}}m + D_{X}S_{k} − S_{k} ∂_{λ}m + m ∂_{λ}S_{k},
n_{(k)} = Y ⊙ S_{k−1} − D_{Q_{k}}n + D_{Y} S_{k} − S_{k} ∂_{λ}n + n ∂_{λ}S_{k}.

Similarly, by (25), there is a symmetric T M-valued (k − 1)-tensor Q_{k−1} such that

X_{(k)} = X ⊙ Q_{k−1} − D_{Q_{k}}X + D_{X}Q_{k} − S_{k} ∂_{λ}X + m ∂_{λ}Q_{k},
Y_{(k)} = Y ⊙ Q_{k−1} − D_{Q_{k}}Y + D_{Y} Q_{k} − S_{k} ∂_{λ}Y + n ∂_{λ}Q_{k},

modulo I_{F}. By (23), the order k symbol of (21) is

D_{X_{(k)}}n + D_{X}(n_{(k)}) + X ⊙ n_{(k−1)} − D_{Y_{(k)}}m − D_{Y} (m_{(k)}) − Y ⊙ m_{(k−1)}

+ m_{(k)} ∂_{λ}n + m ∂_{λ}(n_{(k)}) − n_{(k)} ∂_{λ}m − n ∂_{λ}(m_{(k)}).

Hence, substituting for X_{(k)}, Y_{(k)}, m_{(k)}, n_{(k)}, we have

D_{X⊙Q_{k−1}−D_{Q_{k}} X+D_{X}Q_{k}−S_{k} ∂_{λ}X+m ∂_{λ}Q_{k}}n − D_{Y ⊙Q_{k−1}−D_{Q_{k}} Y +D_{Y} Q_{k}−S_{k} ∂_{λ}Y +n ∂_{λ}Q_{k}}m − D_{Y} (X ⊙ S_{k−1} − D_{Q_{k}}m + D_{X}S_{k} − S_{k} ∂_{λ}m + m ∂_{λ}S_{k})

+ D_{X}(Y ⊙ S_{k−1} − D_{Q_{k}}n + D_{Y} S_{k} − S_{k} ∂_{λ}n + n ∂_{λ}S_{k})
+ (X ⊙ S_{k−1} − D_{Q_{k}}m + D_{X}S_{k} − S_{k} ∂_{λ}m + m ∂_{λ}S_{k}) ∂_{λ}n
− (Y ⊙ S_{k−1} − D_{Q_{k}}n + D_{Y} S_{k} − S_{k} ∂_{λ}n + n ∂_{λ}S_{k}) ∂_{λ}m

− n ∂_{λ}(X ⊙ S_{k−1} − D_{Q_{k}}m + D_{X}S_{k} − S_{k} ∂_{λ}m + m ∂_{λ}S_{k}) + X ⊙ n_{(k−1)} + m ∂_{λ}(Y ⊙ S_{k−1} − D_{Q_{k}}n + D_{Y} S_{k} − S_{k} ∂_{λ}n + n ∂_{λ}S_{k}) − Y ⊙ m_{(k−1)}.

A lot of cancellation now occurs to leave

X ⊙ (n_{(k−1)} + D_{Q_{k−1}}n − D_{Y} S_{k−1} + S_{k−1} ∂_{λ}n − n ∂_{λ}S_{k−1})

− Y ⊙ (m_{(k−1)} + D_{Q_{k−1}}m − D_{X}S_{k−1} + S_{k−1} ∂_{λ}m − m ∂_{λ}S_{k−1})

+ (D_{X}Y − D_{Y} X + m ∂_{λ}Y − n ∂_{λ}X) ⊙ S_{k−1} + (D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m) ∂_{λ}S_{k}

+ D_{D_{X}Y −D_{Y} X+m ∂_{λ}Y −n ∂_{λ}X}S_{k} − (D_{Q_{k}} + S_{k} ∂_{λ})(D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m) and the last two lines vanish modulo I_{F}, which establishes (24) for k′ = N −(p+1) = k−1. We turn now to the order k symbol of (22), which, by (23), is

D_{X_{(k)}}Y + D_{X}(Y_{(k)}) + X ⊙ Y_{(k−1)} − D_{Y_{(k)}}X − D_{Y} (X_{(k)}) − Y ⊙ X_{(k−1)}

+ m_{(k)} ∂_{λ}Y + m ∂_{λ}Y_{(k)} − n_{(k)} ∂_{λ}X − n ∂_{λ}X_{(k)}.

Hence, substituting for X_{(k)}, Y_{(k)}, m_{(k)}, n_{(k)}, we have, modulo I_{F},

0 = D_{X⊙Q_{k−1}−D_{Q_{k}} X+D_{X}Q_{k}−S_{k} ∂_{λ}X+m ∂_{λ}Q_{k}}Y − D_{Y ⊙Q_{k−1}−D_{Q_{k}} Y +D_{Y} Q_{k}−S_{k} ∂_{λ}Y +n ∂_{λ}Q_{k}}X

− D_{Y} (X ⊙ Q_{k−1} − D_{Q_{k}}X + D_{X}Q_{k} − S_{k} ∂_{λ}X + m ∂_{λ}Q_{k}) + D_{X}(Y ⊙ Q_{k−1} − D_{Q_{k}}Y + D_{Y} Q_{k} − S_{k} ∂_{λ}Y + n ∂_{λ}Q_{k}) + (X ⊙ S_{k−1} − D_{Q_{k}}m + D_{X}S_{k} − S_{k} ∂_{λ}m + m ∂_{λ}S_{k}) ∂_{λ}Y − (Y ⊙ S_{k−1} − D_{Q_{k}}n + D_{Y} S_{k} − S_{k} ∂_{λ}n + n ∂_{λ}S_{k}) ∂_{λ}X − n ∂_{λ}(X ⊙ Q_{k−1} − D_{Q_{k}}X + D_{X}Q_{k} − S_{k} ∂_{λ}X + m ∂_{λ}Q_{k}) + X ⊙ Y_{(k−1)} + m ∂_{λ}(Y ⊙ Q_{k−1} − D_{Q_{k}}Y + D_{Y} Q_{k} − S_{k} ∂_{λ}Y + n ∂_{λ}Q_{k}) − Y ⊙ X_{(k−1)}

= X ⊙ (Y_{(k−1)} + D_{Q_{k−1}}Y − D_{Y} Q_{k−1} + S_{k−1} ∂_{λ}Y − n ∂_{λ}Q_{k−1}) − Y ⊙ (X_{(k−1)} + D_{Q_{k−1}}X − D_{X}Q_{k−1} + S_{k−1} ∂_{λ}X − m ∂_{λ}Q_{k−1}) + (D_{X}Y − D_{Y} X + m ∂_{λ}Y − n ∂_{λ}X) ⊙ Q_{k−1} + (D_{X}n − D_{Y} m + m ∂_{λ}n − n ∂_{λ}m) ∂_{λ}Q_{k}, + D_{D_{X}Y −D_{Y} X+m ∂_{λ}Y −n ∂_{λ}X}Q_{k} − (D_{Q_{k}} + S_{k} ∂_{λ})(D_{X}Y − D_{Y} X + m ∂_{λ}Y − n ∂_{λ}X) and the last two lines again vanish modulo I_{F}, so that (25) holds for k′ = N − (p + 1) = k − 1, completing the proof. □

2.2. Proof of Theorem 1. The strategy is to find a dispersionless pair E-equivalent to Π̂ whose integrability condition has minimal order.

We may assume as above that Π̂ is spanned by vector fields D_{X} + m ∂_{λ} and D_{Y} + n ∂_{λ} which commute on shell, where X, Y, m, n depend only on the N-jet of u for some N ∈ ℕ, and that (21)–(22) have order ⩽ k + 1, modulo the ideal I_{F} generated by F and its total derivatives, for ℓ − 1 ⩽ k ⩽ N. By the definition of a dLp, these equations have the form Λ_{1}(F) = 0 and Λ_{2}(F) = 0, where Λ_{1} and Λ_{2} are λ-dependent operators in total derivatives, the latter being T M-valued. In local coordinates we may write Λ_{1} as a finite sum ∑_{α} b_{α}(u_{∞}, λ)D_{α}, and then the symbol of Λ_{1}(F) of any order r ⩾ ℓ + 1 is

Λ_{1}(F)_{(r)} = ∑_{j=0}^{ℓ} ∑_{|α|=r−j} b_{α}(u_{∞}, λ)∂_{α} ⊙ F_{(j)} mod I_{F} .

Since the order r symbol vanishes modulo I_{F} for r ⩾ k + 2, we deduce, starting from r = max{|α| : b_{α} ≠ 0} + ℓ, that b_{α} = 0 mod I_{F} for |α| ⩾ k − ℓ + 2 and that, for k ⩾ ℓ, the order k + 1 symbol has the form L_{1} ⊙ σ_{F} modulo I_{F}, where σ_{F} = F_{(ℓ)} and L_{1} is a symmetric (k −ℓ+1)-vector depending on (u_{∞}, λ); this also holds straightforwardly when k = ℓ − 1. Similarly, for any k ⩾ ℓ − 1, the order k + 1 symbol of Λ_{2}(F) has the form L_{2} ⊙ σ_{F} modulo I_{F} for a T M-valued symmetric (k − ℓ + 1)-vector depending on (u_{∞}, λ).

By Lemma 7, these symbols have the form (24)–(25) modulo I_{F}. Hence, on any solution u and for any θ ∈ Ann(Π), we have L_{1}(θ) ∘ σ_{F} (θ) = 0 and L_{2}(θ) ∘ σ_{F} (θ) = 0 (there is only one independent θ at each point for d = 3 and a pair for d = 4).

For k = ℓ − 1, (L_{1}, L_{2}) is a nonzero (d + 1)-vector-valued function of (u_{∞}, λ). Hence σ_{F} (θ) = 0 for all θ ∈ Ann(Π) and we are done. We may thus induct on k ⩾ ℓ − 1, and suppose that the result holds when (21)–(22) have order ⩽ k. We either have L_{1}(θ) = 0 and L_{2}(θ) = 0 as polynomials in θ ∈ Ann(Π), or that σ_{F} (θ) is not surjective for all such θ. In the latter case, we are done, since the PDE is determined. The former case implies that L_{1} = X ⊙ T_{1} − Y ⊙ U_{1} and L_{2} = X ⊙ T_{2} − Y ⊙ U_{2} for some symmetric (k − ℓ)-vectors T_{1}, U_{1}, T_{2}, U_{2} (the latter pair being T M-valued).

We now let τ_{1}, υ_{1}, τ_{2}, υ_{2} be order k − ℓ operators in total derivatives such that τ_{1}F has order k symbol T_{1} ⊙ σ_{F} modulo I_{F} and so on: concretely, in local coordinates, if T_{1} = ∑_{|α|=k−ℓ} t_{α}(u_{∞}, λ)∂_{α}, we may take τ_{1} = ∑_{|α|=k−ℓ} t_{α}(u_{∞}, λ)D_{α}. We then modify the dispersionless pair by m ↦ m − υ_{1}(F), n ↦ n − τ_{1}(F), X ↦ X − υ_{2}(F), Y ↦ Y − τ_{2}(F). This modification is E-equivalent to Π̂, but the new order k + 1 symbols of (21)–(22) vanish modulo I_{F}, so they have order ⩽ k modulo I_{F}, and the result follows by the inductive hypothesis. □

2.3. Dispersionless pairs characteristic for a quadric. If Π̂ is a dLp for an equation E whose characteristic variety χ^{E} is a quadric, then Π is coisotropic for this quadric by Theorem 1. In this section we investigate the extent to which Π recovers this quadric. We begin with a uniqueness criterion, and then discuss existence. We make essential use of the nondegeneracy conditions (16)–(17), which imply in particular that at each x ∈ M_{u}, the image of Π_{x}: λ ↦ Π_{(x,λ)} does not lie in any proper projective linear subspace of Gr_{2}(T_{x}M_{u}) ⊆ ℙ(∧^{2}T_{x}M_{u}).

Proposition 8. If a 2-plane congruence Π is coisotropic for c_{F}, then for any x ∈ M_{u} and λ ∈ π̂^{−1}(x) at which Π_{x} is an immersion, it is nondegenerate at x. Conversely, at any point x where Π is nondegenerate, there is at most one (quadratic) conformal structure c_{F} on T_{x}M_{u} with Π_{(x,λ)} coisotropic for all λ, and it must be nondegenerate and hyperbolic.

Proof. Suppose first that d = 3, so that Gr_{2}(T_{x}M_{u}) ≅ ℙ(T_{x}^{∗}M_{u}) is a projective plane, and Ann(Π_{x}) is a curve in this plane. If Π is coisotropic, then Ann(Π_{x}) lies on the nonsingular conic {[θ] : σ_{F} (θ) = 0} and so if Π_{x} is immersed, its derivatives of order ⩽ 2 in λ span ℙ(T_{x}^{∗}M_{u}), hence it is nondegenerate. Conversely, two distinct nonsingular conics meet in at most four points, so Ann(Π_{x}) lies on at most one nonsingular conic (which is nonempty, hence hyperbolic), and if Ann(Π_{x}) lies on a singular conic, it lies on a line, hence Π_{x} is degenerate.

Suppose instead that d = 4, so that (the Plücker embedding of) Gr_{2}(T_{x}M_{u}) is the Klein quadric in ℙ(∧^{2}T_{x}M_{u}), and ∧^{2}Π_{x} is a curve in this quadric. If Π is coisotropic, then ∧^{2}Π_{x} lies in a nondegenerate plane section of this quadric, which is a conic: the corresponding lines in ℙ(T_{x}^{∗}M_{u}) belong to one of the rulings of the quadric surface {[θ] : σ_{F} (θ) = 0} in ℙ(T_{x}^{∗}M_{u}). In particular, if Π_{x} is immersed, its tangent does not lie in the quadric, hence it is nondegenerate. Conversely, two distinct nonsingular quadric surfaces meet in a degree four curve (containing at most four lines), so if Π_{x} is nonconstant, it lies on at most one nonsingular quadric surface (which is hyperbolic because it contains lines), and if Π_{x} has image in a singular quadric surface, then the lines pass through a point or lie in a plane, hence ∧^{2}Π_{x} lies in a proper projective linear subspace of Gr_{2}(T_{x}M_{u}), hence Π_{x} is degenerate. □

Proposition 9. Suppose d = 3 and the nondegeneracy condition (16) holds. Then there is a unique conformal structure c for which the 2-plane congruence Π = ⟨X, Y ⟩ is null for all λ if and only if the Monge invariant I(α, β) = 0. This invariant has order 5 in the entries and it distinguishes conics in the projective plane. In the local parametrization with β = λ, this condition is the following (we denote α′ = α_{λ} etc.):

I(α, λ) = 9(α′′)^{2}α^{(5)} − 45α′′α′′′α^{(4)} + 40(α′′′)^{2} = 0.

Suppose d = 4 and the nondegeneracy condition (17) holds. Then there is a unique conformal structure c for which the 2-plane congruence Π = ⟨X, Y ⟩ is (co)isotropic for all λ if and only if the following system of differential equations of order 3 holds, which we write in a partially integrated second order form so (again α′ = α_{λ} etc.)

v′w′′ − v′′w′ = k_{vw}|α′δ′ − β′γ′|^{3/2} for v, w ∈ {α, β, γ, δ},

where k_{vw} are λ-independent and satisfy the “cocycle conditions” k_{vw} + k_{wv} = 0, u′k_{vw} + v′k_{wu} + w′k_{uv} = 0 for u, v, w ∈ {α, β, γ, δ}. In the normalization δ = λ these conditions simplify to: (α, β, γ)′′ = v |α′ − β′γ′|^{3/2}, where v is a λ-independent 3-component vector.

Proof. Let us discuss first the case d = 3. We are looking for a conformal structure c, represented by a pseudo-Riemannian metric g of signature (2, 1), such that the planes Π = ⟨X, Y ⟩ are null. Consider the Pfaffian form θ = dt+α dx+β dy ∈ Ann(Π). The null condition is a single equation c(θ, θ) = 0. Adding to it its λ-derivatives up to order 4, we get a system of 5 equations on 6 coefficients of the metric (5 coefficients if considered up to proportionality). This system is solvable iff (16) holds. Provided this nondegeneracy condition, we can uniquely find c = [g], but in order for it to be supported on M_{u} (and not on M̂_{u}) the ratio of the coefficients of g must be λ-independent. This is equivalent to the condition I(α, β) = 0.

Consider now the case d = 4. Add to the 3 equations c(X, X) = 0, c(X, Y) = 0, c(Y, Y) = 0 their first and second derivatives in λ. The obtained system of 9 equations on 10 coefficients of the metric (9 coefficients if considered up to proportionality) is solvable iff condition (17) holds. Provided this nondegeneracy condition, we can uniquely find c = [g], but in order for it to be supported on M_{u} (and not on M̂_{u}) the ratio of the coefficients of g must be λ-independent. This is equivalent to the system of equations formulated in the proposition. □

3. Projective dependence on the spectral parameter

3.1. Weyl connections and standard dLps. For any 2-plane congruence Π which is characteristic for a bundle of nonsingular quadric hypersurfaces, there is a well-known construction of lifts Π̂_{∇} of Π from Weyl connections ∇, i.e., a torsion-free connections preserving the conformal structure c defining the quadric. Such Weyl connections form an affine space modelled on the vector space of 1-forms on M_{u}.

Lemma 10. Let Π be a nondegenerate 2-plane congruence on M̂_{u} → M_{u}, characteristic for a bundle of quadric hypersurfaces, and ∇ a Weyl connection. Then ∇ induces a connection on M̂_{u} such that the horizontal lift Π̂_{∇} of Π is normal.

Proof. Since ∇ is a conformal connection, it induces a connection on the bundle of coisotropic planes for c, and hence on M̂_{u}, since Π is an immersion. The pullback of ∇ to π̂^{∗}T M_{u} preserves Π and hence, since ∇ is torsion-free, the horizontal lift Π̂ satisfies π̂_{∗}[Π̂, Π̂] = Π. □

We refer to such a lift Π̂_{∇} as a standard dLp. For d = 4, any standard dLp Π̂_{∇} is the unique normal lift of Π = π̂_{∗}(Π̂_{∇}), hence independent of the choice of Weyl connection, as is well known [30]. However, for both d = 3 and d = 4, standard dLps are very special because the connection induced by ∇ on M̂_{u} is projective: M̂_{u} is locally isomorphic to a ℙ^{1}-bundle over M_{u} and if λ is a spectral parameter induced by an affine coordinate on this projective bundle, then horizontal lifts of (λ-independent) vector fields on M_{u} depend quadratically on λ (because vector fields on ℙ^{1} have this form in an affine chart).

Furthermore, with respect to such a projective spectral parameter λ, there is a local parametrization of vector fields spanning Π that is linear in λ, i.e., Π = ⟨V_{1}+λV_{3}, V_{2}+λV_{4}⟩ for λ-independent vector fields V_{i} on M_{u}, so that their lifts are cubic in λ.

When d = 4, these properties follow from the existence of an adapted frame V_{1}, V_{2}, V_{3}, V_{4} for M_{u} such that in the dual coframe θ_{1}, θ_{2}, θ_{3}, θ_{4}, the conformal structure is represented by g = θ_{1}θ_{4} − θ_{2}θ_{3}. Then (up to a choice of orientation) Π = ⟨V_{1} + λV_{3}, V_{2} + λV_{4}⟩, Δ = ⟨V_{1} + λV_{3}, V_{2} + λV_{4}, ∂_{λ}⟩ and it is straightforward to verify that the unique normal lift of Π is

Π̂ = ⟨V_{1} + λV_{3} + m∂_{λ}, V_{2} + λV_{4} + n∂_{λ}⟩,

where the coefficients m, n are given in terms of the structure functions c_{ij}^{k} = θ_{k}([V_{i}, V_{j}]) of the frame as

m = −c_{12}^{4} + λ(c_{23}^{4} − c_{14}^{4} + c_{12}^{2}) − λ^{2}(c_{23}^{2} − c_{14}^{2} + c_{34}^{4}) + λ^{3}c_{34}^{2},
n = c_{12}^{3} − λ(c_{23}^{3} − c_{14}^{3} + c_{12}^{1}) + λ^{2}(c_{23}^{1} − c_{14}^{1} + c_{34}^{3}) − λ^{3}c_{34}^{1}.

These are cubic in λ as required, and compatible with the representation m = m_{1} + λm_{3} and n = m_{2} + λm_{4} for coefficients m_{i} of ∂_{λ} in the lifts of V_{i} that are quadratic in λ.

When d = 3, there is similarly an adapted frame V_{0}, V_{1}, V_{2} on M_{u} with the dual coframe θ_{0}, θ_{1}, θ_{2} such that conformal structure c_{F} is represented by the Lorentzian metric g = 4θ_{0}θ_{2} − θ_{1}^{2} and Π = ⟨V_{0} + λV_{1}, V_{1} + λV_{2}⟩ = ker θ(λ), where

θ(λ) = θ_{2} − λθ_{1} + λ^{2}θ_{0}  (26)

for a (projective) spectral parameter λ. We then have the following fact (cf. [15]).

Lemma 11. Let d = 3 and let Π be as in Lemma 10. Then Weyl connections parametrize projective normal lifts Π̂ of Π.

Proof. By definition any projective lift given by (9), with X = V_{0}+λV_{1}, Y = V_{1}+λV_{2} affine linear, has m, n cubic in λ, i.e., m = ∑_{i=0}^{3} m_{i}λ^{i}, n = ∑_{i=0}^{3} n_{i}λ^{i}. Now Ann(Π) is spanned by the 1-form (26) where θ_{i}(V_{j}) = δ_{ij}. Hence θ(π̂_{∗}[X̂, Ŷ]) = θ([X, Y] − nV_{1} + mV_{2}) is a quartic polynomial in λ determining 5 of the 8 coefficients of m and n. It is straightforward to check that remaining three coefficients are determined uniquely by the Weyl connection (a 1-form has three components at each point). □

3.2. The modified Manakov–Santini master equation in 3D. As mentioned in the introduction, the integrability for a standard dLp Π̂_{∇} in 3D is well-known to be equivalent to the EW equation on (c, ∇) and has the geometric interpretation that any EW manifold locally admits (many) foliations by totally geodesic null surfaces [6, 20] (corresponding to curves in the minitwistor space). We now use this to obtain an alternative derivation of the Manakov–Santini system [25] as a master equation in 3D, or rather a modification of this system which was previously derived in [13] by a different method.

Any totally geodesic null surface has a canonical foliation by null geodesics, so any EW manifold admits a local coordinate system (x, y, t), where x and y are pulled back from local coordinates on the local leaf spaces of a totally geodesic null surface foliation and the induced null geodesic foliation respectively. Thus ∂_{t} is null and orthogonal to ∂_{y} and we can use the freedom in the t coordinate so that the conformal structure has a representative metric

g = 4(dt − b dx)dx − (dy − a dx)^{2}  (27)

for some functions a and b. This has the form 4θ_{0}θ_{2} −θ_{1}^{2}, where θ_{0} = dx, θ_{1} = −dy +a dx, θ_{2} = dt − b dx is the coframe dual to V_{0} = ∂_{x} + a∂_{y} + b∂_{t}, V_{1} = −∂_{y} and V_{2} = ∂_{t}. Thus the null 2-plane congruence Π = ⟨V_{0} + λV_{1}, V_{1} + λV_{2}⟩ is the kernel of

θ(λ) = (dt − b dx) + λ(dy − a dx) + λ^{2}dx = dt + λ dy + (λ^{2} − aλ − b)dx,  (28)

and is equal to W_{0}^{⊥} where

W_{0} = V_{0} +2λV_{1} +λ^{2}V_{2} = ∂_{x} +a∂_{y} +b∂_{t} −2λ∂_{y} +λ^{2}∂_{t} = ∂_{x} +(a−2λ)∂_{y} +(b+λ^{2})∂_{t}. (29)

Since ∂_{y} and ∂_{t} are tangent to level surfaces of x, which are the null surfaces corresponding to λ = ∞, the standard dLp must have the form Π̂_{∇} = ⟨V_{0} + λV_{1} + m′∂_{λ}, V_{1} + λV_{2} + n′∂_{λ}⟩ where m′ and n′ are quadratic in λ.

To obtain a 2-plane congruence in the form (10), we let X = V_{0}+λV_{1}+(a−λ)(V_{1}+λV_{2}) and Y = −(V_{1} + λV_{2}) so that X̂ and Ŷ are given by (9) with m = m′ + (a − λ)n′ and n = −n′. The Lax integrability condition [X̂, Ŷ] = 0 implies n′ is affine linear in λ, while m′ is a quadratic in λ, where the coefficient h of λ^{2} is a function of x and y. We may set h to zero using the coordinate freedom

x ↦ x, y ↦ ρ(x, y), t ↦ ρ_{y}(x, y)^{2}t, λ ↦ ρ_{y}(x, y)(λ − 2ρ_{yy}(x, y)t),

which preserves the form of θ(λ) (hence also g) up to rescaling by ρ_{y}(x, y)^{2} and a redefinition of a and b. The Lax equation now implies that the λ coefficient of m′ differs from −a_{y} by a function of x and y which may be set to zero using the remaining coordinate freedom

x ↦ x, y ↦ y, t ↦ t + τ(x, y), λ ↦ λ − τ_{y}(x, y).

We then find that m′ = −a_{y}λ − b_{y}, n′ = a_{t}λ + b_{t}, and hence

X̂ = ∂_{x} + (−λ^{2} + aλ + b)∂_{t} − ((a_{y}λ + b_{y}) + (λ − a)(a_{t}λ + b_{t}))∂_{λ},
Ŷ = ∂_{y} − λ∂_{t} − (a_{t}λ + b_{t})∂_{λ}.

The Lax integrability condition now reduces to the determined system

(a_{x} − aa_{y} + ba_{t})_{t} = (a_{y} − 2aa_{t})_{y},  (b_{x} − ab_{y} + bb_{t})_{t} = (b_{y} − 2ab_{t})_{y}.  (30)

This is the form of the EW system given in [13, (11)–(12)], except that the x and t variables have been swapped in our conventions and we have used the identity (aa_{y})_{t} = (aa_{t})_{y}. Substituting a = v_{t} and b = u − v_{y} gives the Manakov–Santini system. The modified version (30) may also be written more geometrically as

Δ^{g}a = 0, Δ^{g}b + \frac{3}{2}\{a, b\}_{P} = 0,

where Δ^{g} is the Laplacian of the metric g in (27), and {a, f}_{P} = a_{y}f_{t} −a_{t}f_{y} is the Poisson bracket with respect to the bivector field P = ∂_{y} ∧ ∂_{t} tangent to the null surface foliation.

Remark 3. In [13] a translationally noninvariant version of the MS system was also derived and the question of an explicit equivalence to the standard MS system was raised. However, the translationally noninvariant version is obtained from a generic null surface foliation of the EW manifold, and the coordinate transformation to a totally geodesic null surface foliation will be transcendental in general.

3.3. Arbitrary lifts of 2-plane congruences in 3D. We showed in Proposition 5 that any dLp can be made normal. However, when d = 3, the normal lift of a 2-plane congruence Π is not unique. Instead, the rank 3 distribution Δ = π̂_{∗}^{−1}(Π) ⊆ T M̂_{u} has a unique Cauchy characteristic: a rank 1 subbundle C ⊆ Δ with [C, Δ] = Δ. For a rank 2 subbundle Π̂ ⊆ Δ the normality condition [Π̂, Π̂] ⊆ Δ implies that C ⊆ Π̂, but one generator of Π̂ remains undetermined. In the case of interest that Π = W_{0}^{⊥} is characteristic for a quadric, an easy computation shows that C is spanned by the vector field

Ŵ = W_{0} + σ∂_{λ}, W_{0} := V_{0} + 2λV_{1} + λ^{2}V_{2},

where, using the structure functions c_{ij}^{k} = θ_{k}([V_{i}, V_{j}]) of the adapted frame V_{0}, V_{1}, V_{2}, we have

σ = −c_{01}^{2} + λ(c_{01}^{1} − c_{02}^{2}) − λ^{2}(c_{12}^{2} − c_{02}^{1} + c_{01}^{0}) + λ^{3}(c_{12}^{1} − c_{02}^{0}) − λ^{4}c_{12}^{0}.  (31)

These formulae are compatible with representation σ = m_{0} + 2λm_{1} + λ^{2}m_{2} for the coefficients m_{i} of the lifts of V_{i} which we want to show can be chosen quadratic in λ.

Without loss of generality we may write Π̂ = ⟨Ŵ , Û⟩ with

Û = W_{1} + ψ∂_{λ},  W_{1} = \frac{1}{2}(W_{0})_{λ} = V_{1} + λV_{2}.  (32)

We also write W_{2} = V_{2} = (W_{1})_{λ}. The nondegeneracy condition (15) on Π implies that W_{0}, W_{1}, W_{2} form a (λ-dependent) frame for T M_{u} and indeed

W_{0} ⊙ W_{2} − W_{1}^{2} = V_{0} ⊙ V_{2} − V_{1}^{2}

is the inverse metric to g = 4θ_{0}θ_{2} − θ_{1}^{2}, which is nondegenerate and independent of λ. The Frobenius integrability condition [Π̂, Π̂] = Π̂ is the condition that

[Ŵ , Û] = [V_{0}, V_{1}] + λ[V_{0}, V_{2}] + λ^{2}[V_{1}, V_{2}] + σV_{2} − 2ψ(V_{1} + λV_{2}) + (Ŵ (ψ) − Û(σ))∂_{λ}

is a section of Π̂. Identifying V_{1} ≡ −λV_{2} − ψ∂_{λ}, V_{0} ≡ λ^{2}V_{2} + (2λψ − σ)∂_{λ} modulo Π̂, and assuming that the lift is normal, this reduces to 𝔢 = 0, where

\begin{aligned}𝔢 & := (W_{0} + q_{1} + σ∂_{λ})ψ + 2ψ^{2} − q̂_{0},  q̂_{0} = W_{1} σ + q_{0} σ, \\ q_{1} & = c_{02}^{2} − 2c_{01}^{1} + λ(2c_{12}^{2} − 3c_{02}^{1} + 4c_{01}^{0}) − λ^{2}(4c_{12}^{1} − 5c_{02}^{0}) + 6λ^{3}c_{12}^{0}, \\ q_{0} & = c_{01}^{0} + λc_{02}^{0} + λ^{2}c_{12}^{0}. \end{aligned}  (33)

Using the coefficients of the decomposition [W_{0}, W_{1}] = c̄_{01}^{0}W_{0} + c̄_{01}^{1}W_{1} + c̄_{01}^{2}W_{2} we get

σ = −c̄_{01}^{2},  q_{1} = −c̄_{01}^{1} − σ_{λ},  q_{0} = c̄_{01}^{0}.  (34)

Note that deg_{λ} c̄_{01}^{0} = 2, deg_{λ} c̄_{01}^{1} = 3 and deg_{λ} c̄_{01}^{2} = 4.

Example 5 (dKP). For the dKP equation (2), we have g = dy^{2} − 4dx dt + 4u dx^{2}, θ = dt + λ dy + (λ^{2} − u) dx and

Ŵ = ∂_{x} − 2λ∂_{y} + (λ^{2} + u)∂_{t} + (λu_{t} − u_{y})∂_{λ},  Û = −∂_{y} + λ∂_{t} + ψ∂_{λ},

whence 𝔢 = ψ_{x} − 2λψ_{y} + (λ^{2} + u)ψ_{t} + (λu_{t} − u_{y})ψ_{λ} − u_{yy} + 2λu_{yt} − λ^{2}u_{tt} − ψu_{t} + 2ψ^{2}. In this case, via the change of variables ψ = φ^{−1} + u_{t}, the equation 𝔢 = 0 (33) takes the linear inhomogeneous form

L_{−}(φ) = 2 ⇔ L_{+}(φ^{−1}) = −2φ^{−2},  (35)
where L_{±} = ∂_{x} − 2λ∂_{y} + (λ^{2} + u)∂_{t} + (λu_{t} − u_{y})∂_{λ} ± 3u_{t}.

If we assume ψ either local (= differential) in u or global (= algebraic) in λ, then the only solution is φ^{−1} = 0, implying the existence of a unique dLp of these types.

However, there exist solutions to (35) which are non-algebraic in λ and nonlocal in u. Indeed for any Cauchy data u|_{t=0} that is non-algebraic in λ, we obtain such a solution. In this way we obtain a (characteristic but not projective or local) Lax pair that does not give rise to an EW structure. Moreover, there is no uniqueness for such Lax pairs.

In the following Sections 3.4–3.6 we deduce the cubic behaviour of ψ in λ from the equation 𝔢 = 0, a strengthened nondegeneracy condition, and the requirement that the dLp is local in u. This suffices to establish the projective property, and hence Theorem 2.

3.4. Scalar PDEs in 3D. We first consider the case of a scalar differential equation E : F = 0 of order ℓ, i.e., one PDE (20) on one function u. As before, we assume that the characteristic variety χ^{E} is a quadric, which implies that ℓ is even and the symbol F_{(ℓ)} of the differential operator is a power of a nondegenerate quadratic form: ℓ = 2m, F_{(ℓ)} = Q^{m} for some Q ∈ Γ(S^{2}T M_{u}) on E. (For a second order scalar PDE (1) we get m = 1.) Using the notation of the previous section, we have Q = W_{0} ⊙ W_{2} − W_{1}^{2}.

The order of the conformal structure c_{F} in u satisfies k = ord(c_{F}) ⩽ ℓ, and the strict inequality is possible, for instance, when F is quasilinear (dKP is an example with 0 = k < ℓ = 2). Then the frame and coframe V_{i} and θ_{i} can be chosen to have the same order k in u, while the structure functions c_{ij}^{t} and the coefficient σ in (31) have order ⩽ k + 1.

Let us suppose Π̂ is a normal dLp for E. We want to find an E-equivalent dLp which is projective. Since Π̂ is normal, we may suppose, as in the previous section, that its integrability condition is 𝔢 = 0 with 𝔢 given by (33). Hence by definition of a dLp, 𝔢 = □F for some operator □ in total derivatives. If ψ has order r ⩾ k + 2 then by taking the (r + 1)-symbol of this equation we obtain

W_{0} ⊙ ψ_{(r)} = 𝔢_{(r+1)} = □_{(r+1−ℓ)} ⊙ F_{(ℓ)} = □_{(r+1−2m)} ⊙ (W_{0} ⊙ W_{2} − W_{1}^{2})^{m},

and hence conclude (since W_{0} ⊙ W_{2} − W_{1}^{2} is indivisible by W_{0}) that the symbol of ψ is divisible by that of F. Therefore we can modify ψ off shell (fixed on shell) to obtain an E-equivalent dLp in which the new ψ has order < r. By iterating this process, we may thus assume, up to E-equivalence, that ψ has order ⩽ k + 1 from the outset.

The (k + 2)-symbol of 𝔢 = □F now yields, using equation (33), the relation

W_{0} ⊙ ψ_{(k+1)} − W_{1} ⊙ σ_{(k+1)} = R ⊙ (W_{0} ⊙ W_{2} − W_{1}^{2})^{m}  (36)

for a section R ∈ Γ(S^{k−2m+2}T M_{u}) of the bundle of homogeneous degree k − 2m + 2 polynomials on T^{∗}M_{u}, i.e., R = ∑_{|τ|=k−2m+2} a_{τ}W_{τ}, where we let W_{τ} = W_{j_{1}} ⊙ ⋯ ⊙ W_{j_{t}} for a multi-index τ = (j_{1} ⋯ j_{t}) of length |τ| = t. By modification of ψ and σ off shell, we can bring this function to the form

R = (−1)^{m−1}μ W_{2}^{k−2m+2}.  (37)

Formula (36) then implies that

σ_{(k+1)} = W_{1} ⊙ R ⊙ Q^{m−1} + W_{0} ⊙ T,  ψ_{(k+1)} = W_{2} ⊙ R ⊙ Q^{m−1} + W_{1} ⊙ T (38)

for some T ∈ Γ(S^{k}T M_{u}), and by the normalization (37), the coefficients for R and T are uniquely determined by independent components of σ and hence they are polynomial in λ. In particular, since σ is a quartic polynomial in λ, we conclude that μ ∈ C^{∞}(J^{k+1}M_{u}) is a polynomial in λ with deg_{λ} μ ⩽ 3.

Also, T is a polynomial in λ with deg_{λ} T ⩽ 2. Therefore, ψ_{(k+1)} is a cubic polynomial in λ. Thus, there exists a function ψ_{1} = ψ_{1}(∂^{k+1}u, λ) with deg_{λ} ψ_{1} ⩽ 3 such that ψ_{0} := ψ−ψ_{1} has order ⩽ k in u. Substituting ψ = ψ_{1} + ψ_{0} into the equation 𝔢 = □F we get

(W_{0} + q̃_{1} + σ∂_{λ})ψ_{0} + 2ψ_{0}^{2} = q̃_{0},  (39)

where q̃_{1} = q_{1} + 4ψ_{1}, while q̃_{0} is a expression of order k + 1 in u that we do not write explicitly. However, it follows from (34) that q̃_{1}, q̃_{0} are polynomial in λ with deg_{λ} q̃_{1} ⩽ 3, deg_{λ} q̃_{0} ⩽ 8. We now want to show that ψ_{0} is also polynomial in λ.

In order to do this, it is convenient to carry out computations in the nonholonomic λ-dependent frame (W_{i})_{i=0}^{2} rather than the holonomic frame (∂_{x^{i}})_{i=0}^{2} induced by local coordinates (x^{i})_{i=0}^{2} on M_{u}. Let (a_{i}^{j}) be the transition matrix between these frames and (b_{i}^{j}) be its inverse, i.e., W_{i} = a_{i}^{j}∂_{x^{j}} and ∂_{x^{i}} = b_{i}^{j}W_{j} (summation convention). These vector fields induce vertical vector fields 𝔻_{W_{τ}} on jets via the formulae 𝔻_{W_{τ}} = b_{j_{1}}^{i_{1}} ⋯ b_{j_{t}}^{i_{t}}∂_{u_{i_{1}···i_{t}}}, where τ = (j_{1}, … j_{t}). If ξ is a function on the jet bundle (i.e., a differential operator) with order t symbol ξ_{(t)}, then 𝔻_{W_{τ}} ξ, with |τ| = t, is the the coefficient ξ_{t}^{τ} of ξ_{(t)} in the decomposition ξ_{(t)} = ξ_{t}^{ρ}W_{ρ} (summation over multi-indices ρ with |ρ| = t).

Next, since the dual co-frame to (W_{0}, W_{1}, W_{2}) is (\frac{1}{2}θ_{λλ}, −θ_{λ}, θ), the coefficients on the right hand sides of identities (34) may be written

c̄_{01}^{2} = θ([W_{0}, W_{1}]), c̄_{01}^{1} = −θ_{λ}([W_{0}, W_{1}]), c̄_{01}^{0} = \frac{1}{2}θ_{λλ}([W_{0}, W_{1}]).

This leads to

μ = 𝔻_{W_{1}^{2m−1}W_{2}^{k−2m+2}}(σ) = 𝔻_{W_{1}^{2m−2}W_{2}^{k−2m+3}}(ψ) = θ(𝔻_{W_{1}^{2m−2}W_{2}^{k−2m+2}}(W_{0})). (40)

Identity (38) yields θ(𝔻_{W_{1}^{r}W_{2}^{k−r}}W_{0}) = 0 unless r = 2m − 2. Note that by (38) we have σ_{(k+1)} = μ W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} mod W_{0} ⊙ S^{k}T M_{u}, and this decomposition can be refined:

σ_{(k+1)} = μ W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + γ W_{0} ⊙ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+2} + ⋯
ψ_{(k+1)} = μ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} + γ W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + ⋯,

where γ = 𝔻_{W_{1}^{2m−2}W_{2}^{k−2m+2}}(T) = θ(𝔻_{W_{0}W_{1}^{2m−3}W_{2}^{k−2m+2}}(W_{0})) − θ(𝔻_{W_{1}^{2m−2}W_{2}^{k−2m+2}}(W_{1})) and by dots we mean all terms with other W_{τ} that are irrelevant for the computation. Consequently,

(σ_{λ})_{(k+1)} = (2m − 1)μ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} + (μ_{λ} + 2γ) W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + ⋯

and since σ_{λ} = (θ[W_{1}, W_{0}])_{λ} = θ_{λ}[W_{1}, W_{0}] + θ[W_{2}, W_{0}] = c̄_{01}^{1} − θ[W_{0}, W_{2}] we get

(c̄_{01}^{1})_{(k+1)} = (σ_{λ})_{(k+1)} − μ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} mod W_{0} ⊙ S^{k}T M_{u}.

Thus from (34) we get the following expression for the (k + 1)-symbol

(q̃_{1})_{(k+1)} = −2(σ_{λ})_{(k+1)} + 5μ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} + 4γ W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + ⋯
= (7 − 4m)μ W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} − 2μ_{λ} W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + ⋯

Taking now (k + 1)-symbol of (39), we get

W_{0} ⊙ (ψ_{0})_{(k)} + (q̃_{1})_{(k+1)} ψ_{0} + σ_{(k+1)} (ψ_{0})_{λ} = (q̃_{0})_{(k+1)}.

Denoting (q̃_{0})_{(k+1)} = κ_{0} W_{1}^{2m−2} ⊙ W_{2}^{k−2m+3} + κ_{1} W_{1}^{2m−1} ⊙ W_{2}^{k−2m+2} + ⋯ and extracting the coefficients at the indicated terms (which are unchanged by E-equivalence) we obtain the following system

(7 − 4m)μψ_{0} = κ_{0},  μ∂_{λ}ψ_{0} − 2μ_{λ}ψ_{0} = κ_{1}.

All coefficients of this linear system on ψ_{0} are polynomials in λ. We assume μ ≠ 0 (this condition will be discussed in the next section). Then the first equation uniquely determines ψ_{0}. Moreover, μ divides κ_{0} as a polynomial in λ because otherwise ψ_{0} is a proper rational function and then the second equation, written as (ψ_{0}μ^{−2})_{λ} = κ_{1}μ^{−3}, yields a contradiction.

Thus ψ_{0}, and hence also ψ, are polynomials in λ, and deg_{λ} ψ ⩽ 5. Since the parameter λ is manifestly projective, a projective change should not destroy the polynomial property. Using the special projective transformation λ ↦ λ^{−1} (or a similar projective transformation arbitrarily close to the identity), we conclude that in fact deg_{λ} ψ ⩽ 3.

Moreover, to satisfy smoothness in λ, the function ψ should be compatible with σ in the sense that ψ = m_{1} + λm_{2}, σ = m_{0} + 2λm_{1} + λ^{2}m_{2} for some λ-quadrics m_{i}.

3.5. Nondegeneracy for scalar and vector equations. The λ-dependent quantity μ introduced in (40) characterizes the extent to which the Lax integrability condition depends on the equation E. If this condition does not involve F and its derivatives, the dLp is trivial (holds off shell). We require that the equation shows on the level of the top symbol, i.e., (k + 1)-jet, or equivalently that μ ≠ 0. We first observe that this condition is invariant under admissible transformations of M̂_{u} as a ℙ^{1}-bundle over M_{u}.

Proposition 12. The scalar quantity μ is a relative differential invariant, i.e., transforms by a nonvanishing scalar multiple under admissible transformations.

Proof. The admissible transformations of M̂_{u} have the form (x, λ) ↦ (Φ(x), Ψ(x, λ)), where Φ is a conformal transformation of (M_{u}, [g]) and Ψ(x, λ) = \frac{a(x)+b(x)λ}{c(x)+d(x)λ} is a parametric Möbius transformation. These preserve the algebraic behaviour of the dLp Π̂, and a straightforward computation shows they scale μ by a nonvanishing scalar multiple.

Alternatively, using the framework and normalizations of Section 3.4, σ given by (31) is independent of the adapted frame up to scale and the leading coefficient of its symbol 𝔻_{W_{1}^{2m−1}W_{2}^{k−2m+2}}(σ) is a relative invariant, as required. □

Let us now give the vector version, recalling first the set-up. In this case F: J^{ℓ}(M, V) → W is a determined (nonlinear) differential operator of order ℓ on sections u of a fibre bundle V over M_{u} with values in a rank s vector bundle W. We assume, for simplicity, that V is also a vector bundle of rank s so that we can identify the vertical bundle T_{u}^{v}V along a section u with V. Locally, in coordinates, F has components F^{i} that are scalar differential operators of order ℓ on vector-function u = (u^{j}) of x, where i, j ∈ {1, … s}.

The symbol of F at u = (u^{i}) is a map F_{(ℓ)} : S^{ℓ}T^{∗}M_{u} ⊗ V → W that we identify with an s × s matrix F = (F_{i}^{j}), whose coefficients are polynomials of degree ℓ on T^{∗}M_{u}. Similarly, the symbol of a scalar differential operator φ can be identified with a column in components. The characteristic variety χ^{E} of E is a quadric if det(F) = Q^{m} as before.

The setup of the previous section extends, and μ = θ(𝔻_{W_{1}^{2m−2}W_{2}^{k−2m+2}}(W_{0})) is a section of the bundle V over M_{u} for solutions u of the vector version of (20). The following statement is proved similarly to Proposition 12.

Proposition 13. The section μ is a relative differential invariant, i.e., under admissible transformations it is mapped to another section related to μ by an automorphism of the bundle V. Hence the (non-)vanishing of μ is an invariant property.

Note that μ depends not on a lift or dLp but only on the equation E : F = 0 itself.

Definition 7. In 3D the equation is called nondegenerate (and its dLp nontrivial) if the relative invariant μ is nonzero (identically in λ).

This condition is trivially satisfied if the conformal structure c_{F} has zero order in u, as happens in the dKP case. It can be proved for several classes of PDE in 3D (with ord_{u}(c_{F}) > 0), and we do not know of any integrable equation violating this condition.

Remark 4. In fact, the Manakov–Santini equation, which by [13] is the master equation for EW geometry, is nondegenerate in the sense of this definition. We check this for the modified version, with W_{0} given by (29). Since order of the conformal structure in this formalism is k = 0, and also m = 1, we compute the symbol of W_{0} by (a, b) as (∂_{t}, ∂_{y}) and applying θ given by (28) we get μ = (λ, 1) ≠ 0. Thus the MS equation is nondegenerate and we adopt this condition for our main result.

3.6. Proof of Theorem 2. Any nondegenerate dLp Π̂ is E-equivalent to a normal dLp by Proposition 5. When d = 4, the normal dLp has the projective property, while for d = 3, the main task is to show that, up to E-equivalence, we may assume that ψ in (32) is cubic in λ. The proof almost directly generalizes the scalar version of Section 3.4, so we only indicate important differences on each step.

(i) We begin with equation 𝔢 = □F and, as before, by an off shell modification can arrange that ord ψ ⩽ k + 1, where k is the order of the conformal structure c_{F}. Then its (k + 2)-symbol and (33) yield the following matrix equation

[\begin{matrix}ψ^{1} & σ^{1} \\ ⋮ & ⋮ \\ ψ^{s} & σ^{s} \end{matrix}] ⊙ [\begin{matrix}W_{0} \\ −W_{1} \end{matrix}] = [\begin{matrix}F_{1}^{1} & ⋯ & F_{s}^{1} \\ ⋮ & ... & ⋮ \\ F_{1}^{s} & ⋯ & F_{s}^{s} \end{matrix}] ⊙ [\begin{matrix}R^{1} \\ ⋯ \\ R^{s} \end{matrix}]  (41)

where ψ^{i} are components of the symbol ψ_{(k+1)} and similarly for σ, and where R^{i} are the symbols of some operators in total derivatives. Multiplying this equation by the adjugate matrix adj(F) (which satisfies adj(F)F = det(F)I = Q^{m}I) and denoting the rows of the resulting left-hand side matrix [ψ̃^{i} σ̃^{i}] we get the equations

W_{0} ⊙ ψ̃^{i} − W_{1} ⊙ σ̃^{i} = R_{i} ⊙ (W_{0} ⊙ W_{2} − W_{1}^{2})^{m}, i ∈ {1, … s},

from which we obtain a vector analogue of equation (38) for each component i ∈ {1, … s}. Moreover, we can obtain normalization analogues R_{i} = (−1)^{m−1}μ_{i} W_{2}^{k−2m+2} of (37). This implies that ψ̃^{i} and hence ψ^{i} can be chosen polynomial in λ, moreover deg_{λ} ψ^{i} ⩽ 3. (ii) Thus there exist a decomposition ψ = ψ_{1} + ψ_{0}, where ψ_{1} is at most cubic in λ and has (k + 1)-symbol (ψ^{i}) at u = (u^{j}), while ψ_{0} has order ⩽ k. Substituting this into the constraint 𝔢 = □F we obtain a vector analogue of equation (39). Taking its (k+1)-symbol and applying adj(F) again gives

q̃_{1}^{i}ψ_{0} + σ̃^{i}∂_{λ}ψ_{0} = q̃_{0}^{i}, i ∈ {1, … s}.

If μ = (μ_{1}, … μ_{s}) is nonzero (identically in λ), we conclude by the same argument that ψ_{0} is polynomial in λ with deg ψ_{0} ⩽ 5. Hence ψ is a polynomial in λ with deg ψ ⩽ 5. (iii) Finally, if the coefficients of ψ at λ^{4} or λ^{5} are nonzero, then a Möbius transformation λ ↦ \frac{a(x)+b(x)λ}{c(x)+d(x)λ} arbitrary close to the identity maps the system of vector fields Ŵ and Û = V_{1}+λV_{2}+ψ∂_{λ} (after taking a proper linear combination and clearing denominators) to a system of the same form with a new ψ of higher degree. Thus we must have deg_{λ} ψ ⩽ 3, which is a projectively invariant property. In addition to Û the vector field Ŵ − λÛ = V_{0} + λV_{1} + (σ − λψ)∂_{λ} have degree ⩽ 3 in λ. Indeed, under a change of the adapted frame (V_{0}, V_{1}, V_{2}) and a projective change of parameter λ this field becomes of the form Û and so the claim follows from (i)–(iii).

Theorem 2 is now immediate. By Lemma 11 normal lifts with this projective property are bijective with Weyl connections for d = 3, while for d = 4 the normal lift is unique by Lemma 4. Thus for d = 3 or d = 4, the standard Lax pair of (M_{u}, c_{F} , ∇) or (M_{u}, c_{F}) is E-equivalent to Π̂. □

4. Applications and generalizations

4.1. Pseudopotentials. In this paper we have defined dispersionless integrable systems using a Lax pair of vector fields. In 3D, an alternative approach relies instead on pseudopotentials or nonlinear dispersionless Lax pairs, cf. [39, 28, 18].

Definition 8. A pseudopotential for a PDE F = 0 is a function S: M_{u} → ℝ whose derivative dS satisfies an overdetermined system of two equations that are compatible on shell, i.e., when F (j^{ℓ}u) = 0.

Locally, in coordinates (x, y, t), we may write these equations as S_{x} = A(S_{t}) and S_{y} = B(S_{t}) where A and B also depend on (x, y, t). If they depend on (x, y, t) (only or also) through a section v of a vector bundle over M_{u}, and the integrability condition ∂_{y}(A(S_{t})) = ∂_{x}(B(S_{t})) is required to hold identically in S_{t}, we obtain a PDE system on v. Dispersionless integrable systems are often defined as those determined PDEs arising in this way.

More invariantly, the two equations determine a codimension two (hence 4-dimensional) submanifold N of the cotangent bundle T^{∗}M_{u} and S is a pseudopotential with respect to these equations if dS takes values in N. The integrability condition means that N is coisotropic for the canonical symplectic form Ω on T^{∗}M_{u}. Here we recall that Ω = dτ where τ is the tautological 1-form on T^{∗}M_{u} (with β^{∗}τ = β for any 1-form β on M_{u}). The coisotropic condition means that the pullback of Ω to N has rank two, hence a 2-dimensional radical (or kernel).

Locally N is a fibre bundle over M_{u} and we may take λ = S_{t} as a fibre coordinate in the above explicit formulation. Thus

∂_{y}(A(S_{t})) = A_{y} + A_{λ}S_{tx} = A_{y} + A_{λ}B_{t} + A_{λ}B_{λ}S_{tt},
∂_{x}(B(S_{t})) = B_{x} + A_{λ}S_{ty} = B_{x} + A_{t}B_{λ} + A_{λ}B_{λ}S_{tt},

and so the integrability condition is

A_{y} − B_{x} = {A, B}_{P} := A_{t}B_{λ} − A_{λ}B_{t},  (42)

where {A, B}_{P} is the Poisson bracket of A and B with respect to the Poisson structure P := ∂_{t} ∧ ∂_{λ}; equivalently, the vector fields ∂_{x} + P (dA) and ∂_{y} + P (dB) commute, where P (dA) = A_{t}∂_{λ} − A_{λ}∂_{t} and P (dB) = B_{t}∂_{λ} − B_{λ}∂_{t} are the hamiltonian vector fields associated to A and B by the Poisson structure P (cf. e.g. [14]).

Alternatively, if (42) holds, then the pullback of Ω to N is

dλ ∧ dt + (A_{λ}dλ + A_{t}dt) ∧ dx + (B_{λ}dλ + B_{t}dt) ∧ dy − (A_{t}B_{λ} − A_{λ}B_{t})dx ∧ dy = (dλ − A_{t}dx − B_{t}dy) ∧ (dt + A_{λ}dx + B_{λ}dy)

and its radical is the dLp spanned by ∂_{x} + P (dA) and ∂_{y} + P (dB).

Conversely, let Π̂ be a dLp on π̂: M̂_{u} → M_{u}. On shell, Π̂ is integrable and so M̂ fibres locally over a minitwistor space Tw [20]. At least locally Tw admits a nondegenerate (and necessarily closed) 2-form (such as dz_{1} ∧ dz_{2} in local coordinates); this then pulls back to a closed 2-form ω on M̂_{u} with radical Π̂. We may therefore write (locally, on shell) ω = dα for a 1-form α on M̂_{u}, which we may assume vanishes on the fibres of M̂_{u} over M_{u}; hence we may write α = (Id, α̃) ∘ π̂_{∗} for a section (Id, α̃) of π̂^{∗}T^{∗}M_{u} = {(p̂, ξ) ∈ M̂_{u} × T^{∗}M_{u} | ξ ∈ T_{π̂(p̂)}^{∗}M_{u}}. Then α̃: M̂_{u} → T^{∗}M_{u} is an immersion whose image is coisotropic, since α̃^{∗}τ = α and so α̃^{∗}Ω = dα = ω has rank two with radical Π̂.

In order to do this off shell, we have to work modulo the PDE system. However, the construction of the coisotropic immersion α̃ from a dLp requires integration, and so it may be necessary to pass to a covering system.

Example 6 (dKP). We illustrate this with the well-known example of the dKP equation (2) (u_{x} + uu_{t})_{t} = u_{yy} with dLp (3). We must now find a function f so that ω = fη ∧ θ is closed modulo the equation, and then a 1-form α such that dα = ω modulo the equation. For the first step, it happens in this case that f = 1 works. For the second, setting

α = (\frac{1}{3}λ^{3} − uλ − v)dx + (\frac{1}{2}λ^{2} − u)dy + λ dt,

we have that dα = η ∧ θ modulo the covering system v_{t} = u_{y} and v_{y} = u_{x} + uu_{t}. Thus the pseudopotential system is S_{x} = \frac{1}{3}S_{t}^{3} − uS_{t} − v and S_{y} = \frac{1}{2}S_{t}^{2} − u.

Note that the above nonlocality (usage of v) may be avoided by using the potential form u_{xt} + u_{t}u_{tt} − u_{yy} = 0 of dKP. In this case the pseudopotential S is given by the equations: S_{x} = λ^{3}/3 − u_{t}λ − u_{y}, S_{y} = λ^{2}/2 − u_{t}, S_{t} = λ. In both cases the parameter λ is aligned to the Lax pair in the sense that it is the projective parameter on the correspondence bundle M̂_{u} → M_{u}. This is no longer so with Manakov–Santini system (6).

Example 7 (MS). The MS system does admit a pseudopotential formulation; however it is neither local in u, v nor rational in λ. The system

σ(R_{x} − P_{t}) = (u_{t}λ + u_{y})(Q_{x} − P_{y}), σ(R_{y} − Q_{t}) = u_{t}(Q_{x} − P_{y}),  (43)
σP_{λ} = (λ^{2} + v_{t}λ − u + v_{y})(Q_{x} − P_{y}), σQ_{λ} = (λ + v_{t})(Q_{x} − P_{y}), σR_{λ} = (Q_{x} − P_{y}) (44)

with σ = u_{y}λ + uu_{t} − u_{t}v_{y} + u_{y}v_{t}, is a differential covering, meaning it is compatible modulo MS. Here the last three equations (44) determine the behaviour in the spectral parameter λ, while the first two equations (43) yield a pseudopotential S via the system S_{x} = P, S_{y} = Q, S_{t} = R. Indeed, one can verify that the differential ω = dα of the 1-form α = P dx + Qdy + Rdt on M̂_{u} satisfies X̂⌟ ω = Ŷ⌟ ω = 0 modulo MS and (43)–(44), where X̂ = X̃|_{λ̃=λ}, Ŷ = Ỹ |_{λ̃=λ} in terms of formula (8) are vector fields on M̂_{u} forming the Lax pair (with parameter λ projective).

4.2. Twistor interpretation via contact coverings. To relate the pseudopotential formulation more closely to the dLp formulation, we focus on the first order quasilinear system for sections of π̂: M̂_{u} → M_{u} which correspond to hypersurfaces in the twistor space. We refer to this PDE system as a contact covering of E because the equation it defines is a codimension 2 submanifold Q of J^{1}π̂ (the bundle of 1-jets of sections of π̂), which is a contact manifold.

This viewpoint gives an alternative way to understand why contact coverings are equivalent to dLps. For this, let α be a contact form on J^{1}π̂ representing the standard contact structure and let α_{Q} be its restriction on Q. Then for ω_{Q} = dα_{Q} we have: α_{Q} ∧ ω_{Q}^{d−1} = 0 on shell, but α_{Q} ∧ ω_{Q}^{d−2} ≢ 0, which implies that α_{Q} has a 2-dimensional radical Π̂ ⊆ T Q: ξ⌟ α_{Q} = 0 and ξ⌟ ω_{Q} = 0 for all ξ ∈ Π̂.

If Q is quasilinear, as we require in the definition of a contact covering, then Π̂ is projectible along the fibres π_{1,0} : Q → J^{0}π̂ = M̂_{u} and so it induces a pushforward distribution of rank 2 in T M̂_{u}, which is a dLp in our formalism. This is how a nonlinear covering induces a linear one, and the inverse relation is given by a lift.

We summarize the observed relations into the following diagram, intertwining the twistor and jet concepts:

\xymatrix{ & (J^{1}π̂)^{2d+1} \ar@{-->}[dl] \ar[dd]_{π_{1}} \ar[dr]^{π_{1,0}} &  & Q^{2d−1} \ar@{^{(}->}[ll] \ar@{.>}[dl] \ar[dr]^{Π̂^{2}} \\ (T^{∗}M_{u})^{2d} \ar[dr] &  & J^{0}π̂ = M̂_{u}^{d+1} \ar[dl]_{ℙ^{1}} \ar@{-->}[dr]^{Π̂^{2}} &  & ℙ(T^{∗}Tw)^{2d−3} \ar[dl]_{ℙ^{d−2}} \\  & M_{u}^{d} &  & _{Tw_{d−1}}}

Here the dotted arrow is the restriction of the jet-projection to the contact covering Q, arrows labelled by Π̂ are (local) quotients by the corresponding foliations, and Tw^{d−1} is the mini-twistor or the twistor space for d = 3 or d = 4 respectively.

The dashed arrow is well-defined locally, when a local coordinate (spectral parameter) on the fibre M̂_{u} is chosen, but it may fail to exist globally with respect to the spectral parameter λ and locally with respect to the dependent variable u. For d = 3, this is precisely the theory of pseudopotentials as discussed in Section 4.1. In this case, the space ℙ(T^{∗}Tw) is the real Penrose twistor space (projecting to the Hitchin mini-twistor space with fibre ℙ^{1}) that embeds into the complex twistor space Tw_{ℂ}, which is the complexification of Tw of the case d = 4, via a conformal Killing reduction [21]. We leave to the reader a specification of relations between different real forms (signatures of the conformal structure—related by Wick rotations in physics language).

When d = 4 an analogue of the theory of pseudopotentials has been developed in [34]. Geometrically, this involves making a further projection to ℙ(T^{∗}M_{u}) on the left hand side of the above diagram (this is why the Lax pairs of [34] are homogeneous in ∂ψ for the covering function ψ). This is a 7-dimensional contact manifold, so two equations suffice to define a 5-dimensional submanifold M̂_{u}. In this formalism the Lax pair is given by contact hamiltonian vector fields.

4.3. Extensions of the theory. First, as noted in the introduction, in 2D, the theory of dispersionless Lax pairs is vacuous, essentially because there is only one 2-plane congruence. However, if we relax the assumption that the Lax pair is transverse to the fibres of M̂_{u} over M_{u}, this objection evaporates. The characteristic condition means that at points of tangency, the projection of the Lax distribution is a characteristic direction. In particular, when the characteristic variety is a quadric (two points), we expect two points of tangency, with the background given by the spinor-vortex equations [2].

Secondly, it would be nice to relax the requirement that the PDE system F: J^{ℓ}(M, V) → W determined in the sense that rank(W) = rank(V). The theory in this paper should at least extend to (formally) overdetermined systems (rank(W) ⩾ rank(V)) which are compatible, so that the characteristic variety is a hypersurface. We would then need to use the compatibility conditions to generalize Theorem 1.

For truly overdetermined systems, with characteristic variety of higher codimension, it would be necessary also to replace Lax pairs by Lax distributions of higher rank. Recently the characteristic property was confirmed in [19] for paraconformal structures generalizing EW structures to higher dimension, and we suggest that it applies universally.

Finally, with the latter idea, the restriction to dimensions d = 3, 4 can be relaxed. This would extend the framework of integrability via geometry to a wider context.

Acknowledgements. We are grateful to Jenya Ferapontov for helpful discussions. In particular, he pointed out that dispersionless Lax pairs for PDEs of Hirota type are always characteristic, suggesting that this might be true more generally, and later drew our attention to difficulties in establishing the projective property for Lax pairs in 3D.

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DC: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. Email: D.M.J.Calderbank@bath.ac.uk

BK: Institute of Mathematics and Statistics, UiT the Arctic University of Norway, Tromsø 90-37, Norway. Email: Boris.Kruglikov@uit.no