Second-order PDEs in 3D with Einstein-Weyl conformal structure

Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the symmetrised Ricci tensor of D vanishes. Three-dimensional Einstein-Weyl structures arise naturally on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, the covector w is a somewhat more mysterious object, recovered from the Einstein-Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge-Ampere type), the covector w is also expressible in terms of the equation, thus providing an efficient dispersionless integrability test. The knowledge of g and w provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein-Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein-Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.


Introduction
We consider second-order partial differential equations (PDEs) in 3D, where u is a scalar function of the three independent variables x 0 , x 1 , x 2 , and we denote u i = u x i , u ij = u x i x j , etc. For every solution of equation (1) the corresponding characteristic variety, i≤j ∂F ∂u ij p i p j = 0, defines a conformal structure g = g ij dx i dx j where the symmetric matrix g ij 3×3 is inverse to the matrix of the symbol g ij 3×3 , with g ij = 1+δ ij 2 ∂F ∂u ij (no summation).
Here and in what follows we assume the nondegeneracy condition det g ij ≡ 0, i.e. [g] is well-defined on a generic solution of (1). Equations with nondegenerate characteristic variety will be called nondegenerate.
We will be interested in equations (1) whose characteristic conformal structure g satisfies the Einstein-Weyl property on every solution of (1) (PDEs with EW property for short). Recall that Einstein-Weyl geometry is defined by a triple (D, g, ω) where D is a symmetric connection, g is a conformal structure and ω is a covector such that [6]: (a) connection D preserves the conformal class [g]: Dg = ωg; (b) trace-free part of the symmetrised Ricci tensor of D vanishes.
In coordinates, this gives where ω = ω k dx k is a covector, D k denotes covariant derivative, R (ij) is the symmetrised Ricci tensor of D, and Λ is some function. In fact one needs to specify g and ω only, then the first set of equations (2) uniquely defines the corresponding Weyl connection D. We will refer to ω as the Weyl covector. It was shown in [21] that, for broad classes of translationally invariant equations (1), the Weyl covector is expressed in terms of g by the explicit formula ω k = 2g kj D x s (g js ) + D x k (ln det g ij ), where D x k denotes the total derivative with respect to x k : We emphasize that in the general case formula (3) is no longer valid. Finding a universally applicable formula for the Weyl covector is one of the main objectives of this paper. Since the characteristic conformal structure g depends on the 2-jet of u, one can show that the Weyl covector depends on no more than 3-jets, and is linear in the third-order partial derivatives of u. (Recall that the k-jet of u at a point x can be identified with the collection of partial derivatives ∂ ν u of order |ν| ≤ k.) We recall that Einstein-Weyl equations (2) are integrable by twistor-theoretic methods [28]; in [12] this was explicitly demonstrated in the Manakov-Santini gauge. PDEs (1) satisfying EW property can be viewed as reductions of the Einstein-Weyl equations. This, in particular, implies the existence of a dispersionless Lax representation [5]. We recall that dispersionless Lax pair consists of two parameter-dependent vector fieldsX,Ŷ for which the integrability condition [X,Ŷ ] ∈ span X ,Ŷ holds identically modulo (1). Relations of dispersionless integrable systems to Einstein-Weyl geometry have been discussed in [35,4,15,16,17,14].

Summary of main results
Partial classification results. In Section 2 we demonstrate that EW property is an efficient classification/integrability criterion. To illustrate the approach we obtain complete lists of PDEs with EW property within the following three classes: • Dispersionless lattice equations u xy = f (x, y, t, u, u x , u y , u t , u tt ).
Modulo natural equivalence transformations preserving this class there exists a unique example with f utt,utt = 0, the so-called Boyer-Finley equation u xy = e utt [3]. This example shows that EW property is a rather stringent constraint.
• Nonlinear wave equations u tt = f (x, y, t, u, u x , u y , u t ) u xy .
The EW property leads to a generic case f = sinh 2 ut uxuy , plus a number of degenerations.
The EW property leads to a generic case f = c 2 uxuy cosh 2 ct , plus a number of degenerations.
For all equations arising in the classification we calculate the corresponding Einstein-Weyl structures and dispersionless Lax pairs. The structure of contact symmetry algebras indicates that all resulting equations are contact non-equivalent.
Reconstruction of the Weyl covector. In Section 3 we outline a general procedure to calculate the covector ω. This procedure applies to all equations that are not of Monge-Ampère type, and gives an expression for ω in terms of the equation (see Section 3.1 for Monge-Ampère conditions in 3D). We look for ω in the form where Ω k are given by formula (3), and φ k are the 'correction' terms. Substituting g, ω into Einstein-Weyl conditions (2) and splitting the resulting equations in the third-order derivatives of u, we conclude that the correction terms φ k must be functions of the 2-jet of u only. Furthermore, along with a number of differential relations, φ k must satisfy an algebraic system of 20 linear inhomogeneous equations which, in the non-Monge-Ampère case, determines φ k uniquely. In other words, the system has the form Aφ = B where A is a 20 × 3 matrix of rank = 3, and B is a 20-component vector (both depend on the 2-jet of u). We supply a Mathematica program which calculates the linear system, and the Weyl covector ω. Summarising, we have the following result.
Theorem 1. For every nondegenerate non-Monge-Ampère equation (1) with EW property, the Weyl covector ω is algebraically determined by the equation.
Remark. For Monge-Ampère equations the matrix of the linear system A and the vector B vanish identically, and further analysis is required to reconstruct φ k . In fact, the EW conditions provide an overdetermined differential system for φ which, in generic case, implies a formula for φ through differential closure (compatibility analysis) of the system. We demonstrate this with examples in Section 2.
As a bi-product of our analysis we obtain a remarkable fact that, for any second-order PDE (1) with EW property, 'freezing' the 1-jet of u (that is, giving the variables x i , u, u i arbitrary constant values), results in an integrable Hirota type equation F (u ij ) = 0.
General formula for dispersionless Lax pair. For equation (1) with EW property, in Section 4 we propose an algorithm to calculate the corresponding dispersionless Lax pair. Here is a brief summary. Let g and ω be the characteristic conformal structure and the Weyl covector, respectively. Let us introduce the so-called null coframe θ 0 , θ 1 , θ 2 such that Let V 0 , V 1 , V 2 be the dual frame, and let c k ij be the structure functions defined by commutator The Lax pair is given by vector fieldŝ here ω i are the components of the Weyl covector: ω = ω i θ i . In combination with Theorem 1 we have the following result. This result sounds, in a sense, surprising: intuition coming from the theory of soliton equations tells us that reconstruction of a Lax pair for a given PDE (known to be integrable) should require 'integration' of some kind.
Rigidity conjecture. In Section 5 we formulate a rigidity conjecture which states that, in the non-Monge-Ampère case, every PDE (1) with EW property can be reduced to a dispersionless Hirota form F (u ij ) = 0 via a suitable contact transformation. In other words, all dependence on the 1-jet variables x i , u, u i can be eliminated (for Monge-Ampère equations this is not true).
To illustrate this phenomenon we consider a PDE [32] u tt = u xy u xt for which EW property is equivalent to the Chazy equation ϕ ′′′ + 2ϕϕ ′′ − 3ϕ ′2 = 0. We prove that any deformation of the form which satisfies EW property, is trivial (contact-equivalent to the undeformed equation). We believe that our method of proof can be extended to the general case.

Examples and classification results
Given a class of second-order PDEs in 3D, we impose Einstein-Weyl conditions for the characteristic conformal structure g to obtain classification results. This procedure can be viewed as a 'dispersionless integrability test', and is manifestly contact-invariant. Some illustrative examples are given below. We emphasize that in all examples the Weyl covector ω, as well as the associated dispersionless Lax pair, are expressible in terms of the equation by explicit formulae that work for all special cases arising in the classification.

Dispersionless lattice equations
Here we consider equations of the form In the translationally invariant case, such equations arise as dispersionless limits of integrable lattices see [20]. The characteristic conformal structure of equation (4) has the form Assuming f utt,utt = 0 (which is equivalent to the requirement that equation (4) does not belong to the Monge-Ampère class), one can show that the Weyl covector is given by the following formula in terms of the right-hand side f : where D t denotes the total t-derivative. The requirement that g, ω satisfy Einstein-Weyl conditions on every solution of equation (4) leads to a system of differential constraints (integrability conditions) for the right-hand side f , the simplest of them being where ϕ(t) is an arbitrary function. It can be set equal to zero via a suitable transformation t → a(t), u → b(t)u + c(t), thus leading to the unique canonical form known as the Boyer-Finley (BF) equation [3]. This example demonstrates rigidity of the Einstein-Weyl requirement.
Integrable equations of type (4) possess a Lax representation [X,Ŷ ] ∈ span X ,Ŷ witĥ Remarkably, this Lax pair works modulo integrability conditions satisfied by f and is therefore fully invariant under the equivalence transformations preserving class (4). For BF equation (5) it

Nonlinear wave equations
Here we consider quasilinear equations of the form The characteristic conformal structure is the corresponding Weyl covector is given by At this stage, ϕ(t) is some function to be determined. The requirement that g, ω satisfy Einstein-Weyl conditions on every solution of equation (6) leads to a system of differential constraints (integrability conditions) for the right-hand side f . The simplest of them are as follows: plus four more complicated constraints that involve ϕ(t). One of them is Analysis of these constraints shows that for (nonlinear) integrable equations (6) the coefficient f ut cannot equal zero, and we obtain an explicit formula for ϕ(t) in terms of f : It is a non-trivial corollary of the integrability conditions that the right-hand side of this expression is a function of t only. In any case, we have an explicit formula for ω in terms of the equation. Solving the integrability conditions results in the following generic case: as well as a number of singular strata. Normal forms are achieved modulo equivalence transformations x → η(x), y → ψ(y), u → u+tp(x)+tq(y)+r(x)+s(y), translation of the t-variable, rescaling of u, discrete symmetries x ↔ y and t → −t, and the transformation (x, y, t, u) → (x, y, 1/t, u/t), which all leave the class (6) Table 1: Complete list of (nonlinear) integrable cases.
Cases 1-6 are contact non-equivalent: while some of the symmetry algebras have the same dimensional characteristics, none are isomorphic as follows from the Lie algebra structure. In all cases g is the right extension of an infinite ideal by a Lie algebra of dimension ≤ 3: and s ⋄ is a subalgebra-complement. Thus, both the functional dimension and the number of constants are invariantly defined.
(6) This has the same s ∞ as in case (5), but s ⋄ = R ⋉ R is generated by Equations from Table 1 possess a Lax representation [X,Ŷ ] ∈ span X ,Ŷ witĥ is the same as in the formula for the Weyl covector. Note that this Lax pair works for all cases from Table 1 (upon substitution of the corresponding expression for f ). In fact, one can say more: this Lax pair works modulo the integrability conditions satisfied by f , that is, it is invariant under the equivalence transformations used to obtain cases 1-6.
Remark. Contact symmetry algebra of the BF equation, u xy = e utt , contains 6 functions of 1 variable, therefore, it is not equivalent to any of the items in Table 1. Indeed, for the BF equation we have:

Generalised Dunajski-Tod equations
Here we consider Monge-Ampère equations of the form For f = 4e 2ρt this equation was discussed by Dunajski and Tod in the context of hyper-Kähler metrics with conformal symmetry [16]. The characteristic conformal structure of equation (7) has the form One can show that the Weyl covector can be expressed in terms of the right-hand side f : where R = Dtf f . For f = 4e 2ρt we have R = 2ρ, which results in the Einstein-Weyl structure from [16]. The requirement that g, ω satisfy Einstein-Weyl conditions on every solution of equation (7) leads to a system of differential constraints (integrability conditions) for the right-hand side f . The simplest of them are as follows: plus a number of more complicated constraints. Solving the integrability conditions results in the generic case where c is an arbitrary constant), as well as a number of other strata. Normal forms are obtained modulo the following equivalence transformations: (x, y, t, u) → (η(x), ψ(y), t, u + p(x)e t + q(y)e −t ), rescaling of u, translations of the t-variable, and discrete symmetry (x, y, t, u) → (y, x, −t, u), which all leave the class (7) form-invariant. Thus we obtain the following integrable cases: This list can be reduced further via point transformations as follows.
The final list of integrable cases is summarised below.
# Equation (7) Contact symmetry algebra g  Cases 1-6 are contact non-equivalent: this follows from the structure of their contact symmetry algebras, where we use the notation of Section 2.2.
Items 3 and 6 contain a parameter c which is uniquely characterised by the structure equations. Item 1 also contains a parameter c, yet it does not enter the structure equations. In this case nonequivalence does not follow from the symmetry analysis. Instead we consider (point) transformations inducing an automorphism of the symmetry algebra and preserving the orbit structure. It is easy to see that such transformations, leaving the class of Dunajski-Tod equations (7) form-invariant, are only (x, y, t, u) → (X(x), Y (y), t, ku) and so cannot change c. Thus, the parameter c is essential.

Remark 1.
A comparison between the two tables shows that a possible isomorphism may exist for the following two cases: • Table 1 (1) to Table 2 (1). The symmetry algebras are abstractly isomorphic, yet the corresponding two-dimensional subalgebras [s ⋄ , s ⋄ ] have orbits of dimensions 2 and 1 respectively, hence the items are not equivalent.
• Table 1(3) to Table 2(3) (c = 0). The symmetry algebras are abstractly isomorphic, yet the corresponding infinite-dimensional subalgebras s ′ ∞ have orbits of dimensions 2 and 1 respectively, hence the items are not equivalent.
Thus, all integrable equations from Section 2 (Tables 1-2 and BF equation) are pairwise contact non-equivalent.
Remark 2. For items 2, 3 + , 5, 6(c = 2) of Table 2, the infinite part of the symmetry algebra is not perfect: [s ∞ , s ∞ ] s ∞ . Yet a closer analysis shows that the splitting and the numerical characteristics of Table 2 are invariantly defined.
Remark 3. The generalised Dunajski-Tod equation is quasi-linearisable: the contact transformation Φ(x, y, t, u, u x , u y , u t ) = x, y, maps equation (7) to the quasilinear equation where h = 1 4 Φ * (f ). Equation (9) can be viewed as a deformation of the Bogdanov equation [1]. In the case f = 4e 2ρt considered by Dunajski-Tod [16], equation (9) becomes the integrable PDE studied in [1]: u x u yt − u t u xy = u ρ t u tt . The conformal structure for equation (9) is represented by the metric and the corresponding Weyl covector is given by the formula Note that this Weyl covector satisfies formula (3), i.e. no 'correction' is required, while the covector ω for generalised Dunajski-Tod equation does not satisfy (3), with the 'correction' being given by the second term containing R in (8). This demonstrates contact non-invariance of ω given by (3), while the covector ω = Ω + φ given by Theorem 1 is genuinely contact invariant.
The dispersionless Lax pairs for both generalised Dunajski-Tod (7) and generalised Bogdanov (9) equations can be obtained by the recipe from the proof of Theorem 2. For the former, see Example 3 of §4.2. This implies the Lax pair for the latter via the contact transformation Φ.

Reconstruction of the Weyl covector
We begin by describing the constraints for a PDE to be of Monge-Ampère type.

The Monge-Ampère property
Recall that equation (1) is said to be of Monge-Ampère type if its left-hand side can be represented as a linear combination of minors (of all possible orders) of the Hessian matrix of the function u (with coefficients depending on the 1-jet of u). Let us represent equation (1) in evolutionary form u 00 = f (x 0 , x 1 , x 2 , u, u 0 , u 1 , u 2 , u 01 , u 02 , u 11 , u 12 , u 22 ).
To calculate the Weyl covector, we will need explicit differential constraints for the right-hand side f that are equivalent to the Monge-Ampère property. These have only been known in low dimensions [2,34,11,27]. In full generality, they were obtained recently in [24]. In 3D, the Monge-Ampère conditions consist of two groups of equations for f . First of all, for every i ∈ {1, 2} one has the relations Secondly, for every pair of distinct indices i = j ∈ {1, 2} one has the relations Due to the contact invariance of the Monge-Ampère class, the system of nine relations (12)- (13) is invariant under arbitrary contact transformations.

Proof of Theorem 1
Let us consider a second-order PDE in evolutionary form (11). Note that if a particular equation under study is not evolutionary, it can be brought to evolutionary form via a suitable linear change of the independent variables. It will be convenient to rewrite Einstein-Weyl conditions (2) in terms of the Levi-Civita connection of the conformal structure g (choose any representative of the conformal class): where ∇ denotes covariant differentiation in the Levi-Civita connection of g, and r ij is the corresponding Ricci tensor (which is automatically symmetric), see [29]. Since g depends on the 2-jet of the function u, the Ricci tensor r ij depends on the 4-jet of u. This implies that components ω k must depend on the 3-jet of u, furthermore, the dependence of ω k on the third-order derivatives of u must be affine. Analysis of the dependence of the left-hand side of (14) on the fourth-order derivatives of u suggests a substitution where Ω k is given by formula (3), and the 'correction terms' φ k are some functions to be determined (we will see that they can only depend on the 2-jet of u). Under this substitution equations (14) take the form Let us denote by S the system obtained from (16) by eliminating Λ and restricting the resulting five equations to solutions of PDE (11), that is, reducing the result modulo (11) and its differential prolongation. Equations of system S possess terms of several different types: (a) linear in the fourthorder derivatives of u, (b) quadratic in the third-order derivatives of u, (c) linear in the third-order derivatives of u, and (d) depending on the 2-jet of u only. We will discuss them case-by-case below.
(a) Terms linear in the fourth-order derivatives of u. There are two sources of such terms: expressions r ij + 1 2 (∇ i Ω j + ∇ j Ω i ) and 1 2 (∇ i φ j + ∇ j φ i ). Direct calculation shows that all terms with fourth-order derivatives of u coming from the expressions r ij + 1 2 (∇ i Ω j + ∇ j Ω i ) cancel out. Thus, the only source of such terms are expressions 1 2 (∇ i φ j + ∇ j φ i ), and this implies that φ k must be functions of the 2-jet of u only: φ k = φ k (x 0 , x 1 , x 2 , u, u 0 , u 1 , u 2 , u 01 , u 02 , u 11 , u 12 , u 22 ), recall that u 00 can be eliminated via (11). In other words, ansatz (15) captures the dependence of ω on the third-order derivatives of u. For several classes of (translationally invariant) second-order PDEs the terms φ k vanish identically, however, they are not zero in general. Under conformal rescalings g → λg both covectors ω and Ω transform as ω → ω + d ln λ, Ω → Ω + d ln λ. Thus, the covector φ = φ k dx k is invariant with respect to conformal rescalings.
(b) Terms quadratic in the third-order derivatives of u. Such terms come from the expressions r ij + 1 2 (∇ i Ω j + ∇ j Ω i ) − 1 4 Ω i Ω j , and do not involve φ k . Equating to zero the corresponding coefficients we obtain all third-order partial derivatives of the function f with respect to the variables u 01 , u 02 , u 11 , u 12 , u 22 , which identically coincide with the integrability conditions for Hirota-type equations u 00 = f (u 01 , u 02 , u 11 , u 12 , u 22 ) (17) obtained in [25]. This leads to a somewhat surprising conclusion: taking an integrable equation (11) and 'freezing' the 1-jet of u (that is, giving the variables x 0 , x 1 , x 2 , u, u 0 , u 1 , u 2 arbitrary constant values), we obtain an integrable Hirota type equation. Note that the generic integrable Hirota type equation is a highly transcendental object: it coincides with the equation of the genus three hyperelliptic divisor [10].
(c) Terms linear in the third-order derivatives of u. These terms come from the expressions Each of the five equations of system S has seven terms linear in the third-order derivatives u 011 , u 012 , u 022 , u 111 , u 112 , u 122 , u 222 , recall that we work modulo (11) and its differential prolongation. Equating the corresponding coefficients to zero gives 35 relations involving φ k and their first-order derivatives with respect to u 01 , u 02 , u 11 , u 12 , u 22 . Eliminating the derivatives of φ k we obtain a system of 20 equations which are linear inhomogeneous in φ k (we do not write the equations explicitly due to their complexity). It is exactly at this step that we can determine φ (and hence ω) in terms of the function f . It should be stressed that the linear system of 20 equations for φ k is nontrivial only if equation (11) is not of Monge-Ampère type: in this case the linear system can be represented in matrix form Aφ = B where φ = (φ 0 , φ 1 , φ 2 ) T , B is a vector with 20 components and A is a 20 × 3 matrix, whose coefficients depend linearly on the left-hand sides of the Monge-Ampère conditions (12)- (13). For equations of non-Monge-Ampère type the unknowns φ k can be reconstructed uniquely because A necessarily contains a nonzero 3 × 3 minor. This is equivalent to the condition rank(A) = 3, note that we do not require rk(A|B) = 3 as in the Cramer rule. Indeed, the entire set of EW conditions (more precisely, the differential closure of this system) decomposes into constraints on φ k and equations not containing φ k ; the latter are integrability conditions for (11). Thus, part of the constraints Aφ = B contributes to the integrability conditions for the function f .
(d) Terms depending on the 2-jet of u. For Monge-Ampère equations, both the matrix A and the vector B of the linear system Aφ = B vanish identically. In this case further analysis is required. Constraints of the second order in u involve the derivatives of φ k ; this overdetermined system for φ is not in involution. Generically, the differential closure provides more PDEs that can ultimately lead to algebraic formulae for φ via a finite jet of u. Numerous examples show that this is indeed the case, and that the Weyl covector ω can be reconstructed in terms of the equation even for generic Monge-Ampère equations. However, explicit conditions and demonstration of this is outside the scope of our paper.
This finishes the proof of Theorem 1.
Calculations described above to reconstruct the Weyl covector ω are implemented in a Mathematica program which is attached to this submission.

Examples of computations
To illustrate the general procedure, let us go through steps (a)-(c) for the two particular classes.

Example 1. Let us begin with equations
u tt = f (x, y, t, u, u x , u y , u t , u xy ), the evolutionary form of lattice equations from Section 2.1. The characteristic conformal structure is (set b = u xy ): We will assume f bb = 0, otherwise the equation is of Monge-Ampère type.
Step (a): calculation of Ω using formula (3) gives so that our ansatz for ω is where φ k are functions of the 2-jet of u.
Step (b): here we obtain only one non-trivial equation: Step (c): eliminating the derivatives of φ k with respect to the variables u xx , u xy , u xt , u yy , u yt , we obtain a linear system for φ k (which vanishes identically if f bb = 0, that is, if the original equation is of Monge-Ampère type). In the case f bb = 0 this system gives an explicit formula for φ: thus leading to an explicit formula for the Weyl covector ω. HereD t is the truncated total tderivative (all differentiations are with respect to the 1-jet variables only): We will not continue with step (d): according to Section 2.1, it would lead to a conclusion that any equation (18) with EW property is point-equivalent to the BF equation u tt = ln u xy .
Example 2. Let us consider equations of the form The characteristic conformal structure is (set a = u xx , c = u yy ): We will assume that at least one of the second-order derivatives f aa , f ac , f cc is nonzero, otherwise the

Dispersionless Lax pairs
A background solution is the manifold M = R 3 (x 0 , x 1 , x 2 ) or a domain thereof, equipped with a function u solving (1). We encode it into the symbol M u , which can be viewed as graph(u) ⊂ M ×R, as well as its lift into the jet-space inheriting the geometric structure. Of the latter we emphasise the characteristic variety, which is a projectivisation of the null cone of [g] at every point. This bundle is four-dimensional, called the correspondence spaceM u .
Recall that a dispersionless Lax pair (dLp) can be identified with a rank 2 distributionΠ inM u . The distributionΠ depends on a finite jet of the solution u, and is Frobenius integrable modulo equation (1). The natural projection π :M u → M u has projective fiber P 1 with coordinate λ called the spectral parameter; it parametrises null 2-planes Π of the conformal structure [g] on M u .
It was shown in [5] that modulo equation (1) such Lax pair is unique, coisotropic with respect to the characteristic variety, and the lift Π Π has the projective property. In Lemma 4 of [5] it was proved that the Weyl covector ω uniquely determines the lift (see also Lemma 5 of [5]), however, no explicit formula for the lift was provided. This is what we do below in the proof of Theorem 2.

Proof of Theorem 2
Let X, Y be λ-dependent vector fields generating Π, and let and it remains to show that all first-order derivatives of λ on the right-hand sides of (20) can be eliminated. Let θ ∈ Π ⊥ be a (λ-dependent) annihilator of the 2-plane congruence Π. The condition that the Weyl connection D preserves the field of null cones is where we substitute λ = λ(x) prior to differentiation. This condition gives precisely two linearly independent equations on the 1-jet of λ(x), and these imply that all derivatives of λ on the righthand sides of (20) cancel out, leading to the required formulae for m and n. Given g, ω and following the above scheme, let us derive an explicit formula for dLp. First of all, we choose a (nonholonomic) null coframe θ 0 , θ 1 , θ 2 such that Let V 0 , V 1 , V 2 be the dual frame, and let c k ij be the structure functions defined by the expansions The 2-plane congruence is Π = X = V 0 + λV 1 , Y = V 1 + λV 2 and θ(λ) = θ 2 − λθ 1 + λ 2 θ 0 .

Examples of computations
Below we discuss several examples illustrating the calculations described in the proof.
Example 3. For the generalised Dunajski-Tod equations (7) it is convenient to change the representative of the conformal class as follows: Then ω is changed to the new Weyl covector where θ 0 = dx, θ 1 = α, θ 2 = dy is a null coframe. The dual frame is The coefficients ω i = ω new (V i ) are given by The only nonzero structure functions are thus giving the Lax pairX Below we prove two rigidity-type results, which motivate the above conjecture and demonstrate a technique that may be utilised in its proof in full generality. The main tool is the existence of an open orbit in the solution space of some differential equations with respect to their point symmetry groups.

Rigidity result 1
Let us consider Lagrangians of the form It was shown in [26] that the requirement of integrability (EW property) of the corresponding second-order Euler-Lagrange equation implies that the function ϕ(z) satisfies a fourth-order ODE whose general solution is a modular form of weight one and level three known as the Eisenstein series E 1,3 (z).

Proposition 1. Every Lagrangian of the form
whose Euler-Lagrange equation satisfies EW property, is equivalent to its undeformed version (22) via a change of variables. In other words, Lagrangian (22) is rigid within the class (24). (24) is

Proof. The Euler-Lagrange equation for Lagrangian
The corresponding characteristic conformal structure is Looking for ω in the form (15) and substituting into the Einstein-Weyl conditions we obtain (confirming that ω can be expressed in terms of the equation). Furthermore, we obtain four differential constraints for the function f (t, u, u t ): one of them coincides with the ODE (23) in the variable u t = z, while the other three are more complicated. Utilising GL(2)-invariance of ODE (23) [26], we look for a general solution in the form where ϕ(z) is a generic solution of (23), and α, β, γ, δ, q should be considered as functions of the remaining arguments t, u. Under this ansatz, the ODE (23) (in the variable u t = z) will be automatically satisfied. Direct analysis of the remaining constraints reveals that there exists a common factor p such that (pα) t = (pβ) u and (pγ) t = (pδ) u . Introducing the potentials one can set where q can be reconstructed uniquely up to a constant factor, leading to the following final answer: here h(t, u) and g(t, u) are two arbitrary functions. With any f (t, u, u t ) given by (26), equation (25) possesses EW property. It is a common phenomenon that arbitrary functions occuring in the coefficients of integrable systems can be eliminated by a change of variables. This is exactly the case in our example: introducing the point transformation one can show that the densities transform into each other, thus establishing triviality of deformation (24).

Rigidity result 2
Equations of the form have appeared in the classification of integrable hydrodynamic chains [32], where it was shown that ϕ must satisfy the Chazy equation [8] (set u xx = a): ϕ aaa + 2ϕϕ aa − 3ϕ 2 a = 0.

Proposition 2. Every equation of the form
which satisfies EW property, is equivalent to its undeformed version (27) via a suitable contact transformation. In other words, equation (27) is rigid within the class (29).
Proof. The corresponding characteristic conformal structure is g = u 3 xt dxdy −

General rigidity conjecture
The above arguments can be extended to the general case as follows. Let U be the Hessian matrix of a function u, and let F (U ) = 0 be the Hirota master-equation. We will exploit two facts (we change to complex coefficients for classification reasons): • Generic integrable Hirota-type equations belong to the same open Sp(6, C)-orbit, see [25]; • Freezing 1-jet of u in equation (1) with EW property yields an integrable Hirota-type PDE, see §3.2(b).
Thus, a generic second-order PDE with EW property can be represented in the form where A, B, C, D are 3 × 3 matrices depending on 1-jet variables x i , u, u i , such that the 6 × 6 matrix A B C D belongs to Sp(6, C). Under the substitution (34) part of the Einstein-Weyl conditions will be satisfied identically, leaving a (complicated!) system of differential constraints for A, B, C, D in the 1-jet variables. It remains to prove that these constraints are equivalent to the existence of a contact transformation taking PDE (34) into Hirota form F (U ) = 0. The complexity of the resulting differential constraints is a formidable obstacle in this programme.

Concluding remarks
is not flat on every solution) can be transformed, via a suitable Bäcklund transformation, into a translationally invariant form to which the method of hydrodynamic reductions would already apply. Note that in the latter case contact transformations may not be sufficient.
These relations specify the new independent variables y i uniquely modulo transformations y i → ϕ i (y i ), which are point symmetries of equation (36). Bäcklund transformation (38) can be viewed as a 3D version of reciprocal transformations that are well-studied in 2D.