Integrability via geometry: dispersionless differential equations in three and four dimensions

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
The integrability of dispersionless partial differential equations is now well known to admit a geometric interpretation. Twistor theory gives a framework to visualize this for several types of integrable system [20,19,1], as demonstrated by many examples [21,14,10,25].
Recently, such a relation has been established for several classes of second order equations in 3D and one class in 4D [13]. Namely the following equivalences have been established:

Integrability via hydrodynamic reductions
Dispersionless Lax pair with spectral parameter k s k s + 3 + 3 Integrable background geometry Integrability by the method of hydrodynamic reductions in 2D (also written "1 + 1 dimension") was introduced in [6] and later elaborated in [23]. The corresponding theory in d 3 dimensions was developed in [12], but this method applies only to translation invariant equations. In coordinate-free language, this requires the existence of an Abelian contact symmetry group with d-dimensional generic orbits. Thus this approach to integrability, although constructive, is not universal, and so the upper part of the diagram of equivalences 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 aim of this paper is to prove the bottom equivalence for general second order PDEs in 3D and 4D, where "integrable background geometry" means that the canonical conformal structure of the equation is Einstein-Weyl in 3D and self-dual in 4D (these geometries are "backgrounds" for integrable gauge theories [1]).
for a scalar function u on a connected manifold M, dim M = d, where u = u(x) for an independent variable x = (x i ) on M, and ∂u = (u i ) and ∂ 2 u = (u ij ) denote partial derivatives with respect to x. Let M u denote the manifold M with a chosen 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.
Let σ F be the linearization of F in second derivatives, i.e., Invariantly, σ F defines an element of S 2 T x M u , hence a quadratic form on T * M u , which we call the symbol of F . If we replace F by a function with the same zero-locus, σ F changes by a conformal rescaling along F = 0. Hence the conformal class of σ F along F = 0 is an invariant of E, as is the characteristic variety C E := {[θ] ∈ P(T * 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 of F . This is equivalent to det(σ ij (u)) = 0 for a generic solution of (1). • hyperbolic, i.e., M is complex and F is holomorphic, or M is real, F is smooth and the variety {[θ] ∈ P(T M ⊗ C) | σ F (θ) = 0} of complex characteristics complexifies C E . The nondegeneracy of σ F implies that its inverse is a nondegenerate symmetric bilinear form on T x M u for any (x, u) sufficiently close to a generic point of F = 0. The corresponding conformal structure c F will be central in this paper, as it was in [13]. Hyperbolicity implies that along F = 0, c F is uniquely determined by C E , which is a bundle of quadrics dual to the projectivized null cone of c F .
We now define dispersionless Lax pairs [26] as rank one coverings of M u [24].

Definition 1.
A dispersionless pair of order N is a bundleπ :M u → M u called the correspondence space, whose fibres are connected curves, together with a rank two distribution Π ⊆ TM u such that: • at eachx ∈M u ,Π depends only on the partial derivatives of u at x of order N; •Π is transversal to the fibres ofπ in the sense that Π(x) : Thus Π is a 2-plane congruence: a section Π : A spectral parameter is a local fibre coordinate λ onπ. We say that: •Π is characteristic for E if for any solution u of E, anyx ∈M u , and any θ in the annihilator Ann( More concretely, let X, Y be 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 a dispersionless pair onM u is given by liftsX,Ŷ toM u (i.e., dπ(X) = X, dπ(Ŷ ) = Y ) such that the Frobenius condition [X,Ŷ ] ∈ Γ(Π) holds modulo F = 0.
The characteristic condition means that for each solution u and eachx ∈M u , Π(x) 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. The condition is vacuous for d = 2; it means that Π is tangent to the null cone of c F (i.e., degenerate) for d = 3, or contained in the null cone (i.e., totally isotropic) for d = 4. 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 form a 1-dimensional submanifold of the Grassmannian Gr 2 (T x M). For d = 3 this submanifold is a rational curve (≃ projective line) canonically isomorphic to the conic C E ⊆ P(T * x M). For d = 4, it is a disjoint union of two rational curves corresponding to the so-called α-planes and β-planes in C E .
If (M u ,Π) is characteristic and Π is an immersion, we may identifyM u locally with the P 1 -bundle whose fibre over x ∈ M u is (a connected component of) the coisotropic 2-planes at x. By definition, this bundle has a tautological 2-plane congruence, and any torsion-free conformal connection ∇ on M u (called a Weyl connection) determines a lift of this 2-plane congruence to a distribution on the total space. In 3D, it is well known [3] that this lift 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 ). In 4D, it is well-known [20] that the lift is conformally invariant (independent of ∇) and is integrable if and only if (M u , c F ) is has self-dual (SD) Weyl tensor: Thus it is reasonable to expect, following [13], a relation between integrability of the PDE E and the EW/SD property for c F on any solution, by which we mean that the EW/SD property is a nontrivial differential corollary of E. The need for nontriviality here, and in the definition of a dispersionless Lax pair, is illustrated by the PDE ∆u = f (∂u): this PDE is non-integrable for generic f , but its conformal structure is independent of the solution and is flat, so the EW/SD property holds. We address this issue further in Remark 2.
Remark 1. For d = 4, our theory also applies to analytic hyperbolic PDEs in signature (3, 1) by complexification. However, since W ± c F would then be complex conjugate, Theorem 1 below implies that integrability holds only when c F is conformally-flat on solutions, which is very restrictive. For instance, for the classes considered in [13,5] it implies that E is linearizable.
In Section 1, we show that for Π to define an immersion into the coisotropic planes of a cone over a quadric, a certain nondegeneracy condition is required. Conversely, this condition implies that Π immerses (and we give a computational criterion for the existence of the quadric). In Section 2, we show that the standard EW/SD Lax pair has a certain 'normality' property. The key technical result is Theorem 9 in Section 3, where we show that dispersionless Lax pairs are characteristic. Here we need some jet theory which we have suppressed in the rest of the paper-see Remark 2-but which also allows us to extend the theory from second order scalar equations to arbitrary determined PDE systems (defined by arbitrary differential operators between bundles of the same rank). In this more general setting, as we explain in Section 3, the characteristic variety is the support of the kernel of the symbol of the operator [16,22], which is a hypersurface in P(T * M u ) in the determined case.
These results allow us to state and prove straightforwardly the following theorem.
Theorem 1. Let E : F = 0 be a determined PDE system in 3D or 4D whose characteristic variety C E is a quadric (such as a nondegenerate hyperbolic second order scalar PDE ) and let c F be the corresponding conformal structure. Then the integrability of the PDE by a nondegenerate dispersionless Lax pair is equivalent to 3D: the Einstein-Weyl property for c F on any solution of the PDE ; 4D: the self-duality property for c F on any solution of the PDE.
Proof. First, suppose that c F is Einstein-Weyl (with Weyl connection ∇) on any solution of E for d = 3, or that it is self-dual on any solution of E for d = 4. 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 for any u (not necessarily a solution) c F is defined on an open subset of M u . Letπ :M u → M u be the bundle of null 2-planes for d = 3, or the bundle of α-planes for d = 4. Then ∇ (or any Weyl connection for d = 4) defines a lift of the tautological 2-plane congruence Π ⊆π * T M u to a distributionΠ onM u depending on c F and its first derivatives, hence on derivatives of u of order ℓ + 1 (or ℓ if F is quasilinear), cf. Proposition 4. This standard Lax pairΠ (see Section 2) is a nondegenerate characteristic dispersionless pair whose integrability is equivalent to the Einstein-Weyl property of (M u , c F , ∇) for d = 3 (when its integral surfaces project to totally geodesic null surfaces for (c F , ∇) [3,14]), or the self-duality of (M u , c F ) for d = 4 (when its integral surfaces project to α-surfaces for c F [20]). Thus if the latter are nontrivial differential corollaries of E, so is the former. We show that the same is true for anyΠ ′ E-equivalent toΠ at the end of the proof.
Conversely, supposeΠ ⊆ TM u is a dispersionless Lax pair for E and let Π ⊆π * T M u be the underlying 2-plane congruence. By Theorem 9,Π is characteristic, i.e., when F = 0, Π is a coisotropic 2-plane congruence for the conformal structure c F (and for d = 4 we orient M u so that Π is a congruence of α-planes). Since Π is nondegenerate, it immerses into Gr 2 (T M u ). We may thus assume thatM u is an open subset of the bundle of coisotropic 2-planes for all solutions u, and hence, by choosing fibre coordinates, for all u where σ F is nondegenerate.
When d = 3, there is, for any solution u, a unique Weyl connection ∇ on M u such that Π is the ∇-lift of Π, which depends only on finitely many derivatives of u, and so may be extended to all u where c F is defined. Thus for d = 3 or d = 4, thus the standard Lax pair of (M u , c F , ∇) or (M u , c F ) is E-equivalent toΠ, and so the Einstein-Weyl or self-duality equation is a nontrivial differential corollary of E, as required.
Finally, if a dispersionless pairΠ ′ E-equivalent to the standard Lax pair is a differential corollary of a strictly weaker equation E ′ , then the above converse argument implies that the EW/SD property is also satisfied under this weaker equation.
The above theorem 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 (translationinvariant) equations considered in [13], 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 [17] 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 (ℓ if the PDE is quasilinear) derivatives of u, and satisfies a "normality" condition off-shell which is useful in computations. None of these properties were assumed a priori. We conclude, in Section 4, with some applications and potential generalizations, including a new derivation of the Manakov-Santini system [18,8,9] from Einstein-Weyl geometry using the existence of a totally geodesic null surface foliation.

Quadratic cone condition and nondegeneracy
IfΠ is a characteristic dispersionless pair for an equation E whose characteristic variety is a quadric, then Π is coisotropic for a quadratic cone. In this section we investigate the extent to which Π recovers this quadratic cone. We begin with a uniqueness criterion, and then discuss existence.
In particular, if Π x is not constant, its image does not lie in any proper projective linear subspace of Conversely, if this holds, 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.
. Conversely, two distinct nonsingular conics meet in at most two 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.
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, θ) = 0} in P(T * x M u ). In particular, if Π x is immersed, its tangent does not lie in the quadric. 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 ).
with coordinates x = (x, y, t), any 2-plane congruence has generators of the form where the functions α, β depend on (x, y, t) (also through u = u(x)) and the spectral parameter λ. In 4D, with coordinates x = (x, y, z, t), we similarly have where the functions α, β, γ, δ depend on (x, y, z, t) (also through u = u(x)) and the spectral parameter λ. The nondegeneracy conditions may now be written explicitly as: Proposition 3. Suppose d = 3 and the nondegeneracy condition (5) 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.): Suppose d = 4 and the nondegeneracy condition (6) 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 where k vw are λ-independent and satisfy the "cocycle conditions" 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 (5) holds. Provided this nondegeneracy condition, we can uniquely find c = [g], but in order for it to be supported on M u (and not onM 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 (6) holds. Provided this nondegeneracy condition, we can uniquely find c = [g], but in order for it to be supported on M u (and not onM u ) the ratio of the coefficients of g must be λ-independent. This is equivalent to the system of equations formulated in the proposition.

Lifts and normality
The passage from a 2-plane congruence Π to a dispersionless pairΠ can be understood as a lift, with respect to the projectionπ :M u → M u . We writeΠ as the span of vector fieldŝ where X, Y are given by (3)  For any 2-plane congruence Π which is quadratic for a conformal structure c, there is a well-known construction of liftsΠ of Π from Weyl connections, i.e., a torsion-free connections ∇ preserving the conformal structure c. We recall that the Weyl connections form an affine space modelled on the vector space of 1-forms on M.
Proof. Since ∇ is a conformal connection, it induces a connection on the bundle of coisotropic planes for c, and hence onM , since Π is an immersion. The pullback of ∇ toπ * T M u preserves Π and hence, since ∇ is torsion-free, the horizontal liftΠ satisfiesπ * ∂Π(x, λ) = Π(x, λ).
Definition 3. We say that the dispersionless pairΠ ⊆ TM u is normal if the derived distribution ∂Π is tangent to the fibres of the projectionπ for general u (off-shell). In other words, for everyx = (x, λ) ∈M u we haveπ * (∂Π(x)) = Π(x, λ).
Thus if the dispersionless pair is normal, the only integrability condition is the vanishing of the vertical direction of the commutator [X,Ŷ ] modΠ, and this is (a differential corollary) of E. Computationally, it is the ∂ λ coefficient [X,Ŷ ] · λ of the vector field [X,Ŷ ]. Proof. The assumptions imply there is a local parametrization of vector fields spanning Π that is linear in λ: [10]). Then due to the projective structure in the fibres ofπ the lift (7) has m, n cubic in λ (the lift for V i has vertical part quadratic in λ), i.e., m = 3 is a quartic polynomial in λ determining 5 of the 8 coefficients of m and n. The remaining three coefficients are determined uniquely by the Weyl connection (a 1-form has three components at each point).
For d = 4, it is well-known that the lift of Proposition 4 is independent of the choice of Weyl connection [20]. However, the normality condition provides a simple characterization of this lift, and the uniqueness holds even without the quadratic cone condition.
We shall see later, in Proposition 10, that any dispersionless pair can be made normal. order k iff it is a function of the x i and u α with |α| k. 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

Jets and the characteristic condition
The top degree term f (k) , called the (order k) symbol of f , is independent of coordinates and vanishes iff f has degree k − 1 (in which case the order k − 1 symbol is defined). 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 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 (8): u is a solution of (8) iff j ∞ u is a section of E. Then σ F := F (ℓ) , whenever nonzero, is (the pullback to J ∞ M of) the symbol of F in the usual sense.

Remark 2.
In the rest of the paper, we have avoided the jet formalism by using the philosophy [16,24] that a differential equation E ⊆ J ∞ M is a generalized manifold whose "points" are solutions u (or, rather, M u = (j ∞ u)(M) ⊆ E which 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. Working explicitly with jets also clarifies the notion of a differential corollary of an equation F = 0: it is simply a subset of I F (or, more invariantly, the ideal I I F generated by this subset and its total derivatives of arbitrary order). This raises an issue which touched on only briefly in the introduction. We shall see in Section 4 that it is too restrictive in examples to require that the integrability conditions for a dispersionless pair generate I F . Furthermore, there is a freedom to replace a dispersionless pairΠ by an E-equivalent one, and we exploit this in Theorem 1. This may change the ideal I I F that its integrability conditions generate.
is not injective} [22]. If dim V = dim W , then [θ] is characteristic iff σ F (θ) is not surjective. We take dim V = dim W as our definition of a determined system, although the more proper definition is that the characteristic variety is a hypersurface.

Lemma 7.
If D X q − D Y p has order k, for functions p, q on J ∞ (M, V) and sections X, Y of π * ∞ T M, then its order k symbol is 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 (9) 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 .
In the jet formalism, a dispersionless pairΠ lives on a rank 1-bundleπ :M → J ∞ (M, V) (so thatM u = (j ∞ u) * M ) and we letπ ∞ = π ∞ •π :M → M. A 2-plane congruence Π is then a rank 2 subbundle ofπ * ∞ T M, andΠ is a lift of Π to TM. In practice we use a spectral parameter λ to trivializeM over J ∞ (M, V). Then TM is the direct sum of the vertical bundle ofπ, spanned by ∂ λ , andπ * T J ∞ (M, V). Thus if Π is spanned by X, Y ∈π * ∞ T M, we may write the dispersionless pairΠ as the span ofX = D X + m ∂ λ andŶ = D Y + n ∂ λ , where D X and D Y are total derivatives (depending also on λ) and m, n are functions onM. Then 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 We thus have a dispersionless Lax pair 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 8. Let (10)-(11) 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 (10) and (11) are respectively Proof. Suppose that X, Y, m, n depend only on the N-jet of u for some N ∈ N, so that (10)- (11) 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 (10) and (11) are simply (9), 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 (10)-(11) have order k modulo I F . Then (10) 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 (12). Hence Lemma 7 produces a symmetric (k − 1)-tensor S k−1 such that, modulo I F , Similarly, by (13), there is a symmetric T M-valued (k − 1)-tensor Q k−1 such that (9), the order k symbol of (10) is Hence, substituting for X (k) , Y (k) , m (k) , n (k) , we have A lot of cancellation now occurs to leave and the last two lines vanish modulo I F , which establishes (12) for k ′ = N − (p + 1) = k − 1.
We turn now to the order k symbol of (11), which, by (9), is Hence, substituting for X (k) , Y (k) , m (k) , n (k) , we have, modulo I F , and the last two lines again vanish modulo I F , so that (13) holds for k ′ = N − (p + 1) = k − 1, completing the proof.
Theorem 9. Let F = 0 be a (system of ) determined PDEs of order ℓ on a manifold M, i.e., F is a differential operator of order ℓ between sections of bundles of the same rank. Then any dispersionless Lax pair (M ,Π) for F = 0 is characteristic.
Proof. Our strategy is to find a dispersionless pair equivalent toΠ whose integrability condition is as simple as possible, i.e., if necessary, we modify theΠ off-shell (keeping it unchanged on-shell) to control the symbol of its Frobenius integrability constraint. We may assume as above thatΠ 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 (10)-(11) 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 dispersionless Lax pair, 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 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).

Applications and generalizations
Theorem 1 shows that the EW and SD equations are master equations, in 3D and 4D respectively, for determined PDE systems whose characteristic variety is a quadric. For instance, it covers the systems of two first order PDE on two unknown functions considered in [5] (partially re-proving the results from this reference), as well as 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. We therefore illustrate the theory by presenting a determined form for the EW equations, following [9].
For this note that any EW manifold M locally admits a foliation by totally geodesic null surfaces [3,14] (such foliations correspond to curves in the minitwistor space) and any totally geodesic null surface has a canonical foliation by null geodesics. Introduce a local coordinate system (x, y, t) on M, where x and y are pulled back from local coordinates on the local leaf spaces of the null surface foliation and the 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 for some functions a and b. Then for any spectral parameter λ, ∂ x + a∂ y + b∂ t − 2λ∂ y + λ 2 ∂ t is null and its orthogonal 2-plane congruence is the kernel of θ = (dt−b dx)+λ(dy−a dx)+λ 2 dx, which is spanned by X ′ = ∂ x + a∂ y + b∂ t − λ∂ y and Y = ∂ y − λ∂ t . Since ∂ y and ∂ t are tangent to the level surfaces of x, which is the null surface foliation corresponding to λ = ∞, the lifts of X ′ and Y have the form X ′ + m ′ ∂ λ andŶ = Y + n∂ λ where m ′ and n are quadratic in λ. Taking X = X ′ +(λ−a)Y we obtain a 2-plane congruence in the form (3), andX = X +m∂ λ with m = m ′ + (λ − a)n. The Lax equation [X,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 which preserves the form of θ (hence also g) up to rescaling by ρ y (x, y) 2 and 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 We then find that m ′ = −a y λ − b y , n = −a t λ − b t , and the Lax equation reduces to the determined system a, b ∈ ker( + D), where = (∂ x + a∂ y + b∂ t )∂ t − ∂ 2 y , D = (2a y + b t )∂ t − a t ∂ y . (15) More invariantly ( + D)(f ) = ∆ g f + 3 2 {a, f }, where ∆ g is the Laplacian of the metric g in (14), and {a, f } = a y f t − a t f y is the Poisson bracket with respect to the bivector field ∂ y ∧ ∂ t tangent to the null surface foliation. This system is equivalent to the form of the EW equations given in [9, (11)- (12)], except that the x and t variables have been swapped in our conventions. Substituting a = v t and b = u − v y , the EW system (15) can be rewritten as where φ yy (x, y) = ψ(y, t), we obtain the system F = 0, G = 0, known as the Manakov-Santini system [18,9]. However, since it is only equivalent to the EW system up to the gauge freedom in (u, v), its Lax pair is not normal, but is related to the normal EW Lax pair bŷ The Manakov-Santini system illustrates the phenomenon that the EW equations for the canonical conformal structure of a dispersionless integrable system can only be expected to be a differential corollary of the original system in general. Two special cases are worth noting: when a = 0 (i.e., without loss, v = 0) the system for b = u reduces to the dKP equation u xt + (uu t ) t − u yy = 0 (here G = 0 and so the Lax pairs coincide-these EW spaces were introduced in [10]); when u = 0 the Manakov-Santini Lax pair (16) has no derivatives with respect to the spectral parameter, and so the EW space is hyperCR [1,2,7].
There are many other examples in 3D, such as the Lax pairs arising in the central quadric ansatz [11] and for EW spaces in diagonal coordinates [9]. Similarly in 4D, there are Lax pairs which have no derivatives with respect to the spectral parameter (hence cannot be normal), such as the hypercomplex Lax pair in [9] or the Lax pairs for Monge-Ampère equations of Hirota type [4]. Nevertheless, there is a determined 4D master equation [9] which is normal. In general, we have the following observation.
Proposition 10. A nondegenerate Lax pairΠ = X ,Ŷ ⊆ TM u can be made normal by a modification off-shell that is fixed on-shell (on the solutions).
In 4D the Lax pair condition (on-shell) implies similarly dz • π * [X,Ŷ ] = 1 F, dt • π * [X,Ŷ ] = 2 F for some operators 1 , 2 in total derivatives. Let us modifyX =X + A(F )∂ λ ,Ỹ = Y +B(F )∂ λ , where A, B are operators in total derivatives to be determined (they also depend on λ). The new commutation equation is 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 (6).
Hence normal Lax pairs are essentially always available, and provide a canonical choice in 4D, while in 3D, they are determined by a choice of Weyl connection.
We end with remarks on extensions of the theory. First, we noted that the 2D theory of dispersionless Lax pairs is vacuous, essentially because there is only one 2-plane congruence in 2D. However, if we relax the assumption that the Lax pair is transverse to the fibres ofM over M, 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, and the role of the EW and SD equations should be replaced by the spinor-vortex equations [1].
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 9. For truly overdetermined systems, where the characteristic variety has higher codimension, it would be necessary also to replace Lax pairs by Lax distributions of higher rank.