Second-order integrable Lagrangians and WDVV equations

We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form ∫f(uxx,uxy,uyy)dxdy.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$\end{document}By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.


Introduction and summary of the main results
We investigate second-order Lagrangians f (u x x , u xy , u yy ) dxdy (1) such that the corresponding Euler-Lagrange equations, are integrable (in the sense to be explained below). We work in the context of the formal calculus of variations and understand (1) as a formal action functional which generates Eq. (2) via the Euler-Lagrange operator applied to the Lagrangian density f . Examples of integrable Lagrangians (1) have appeared in the mathematical physics literature; thus, the Lagrangian density governs integrable geodesic flows on a 2-torus which possess a fourth-order integral polynomial in the momenta [3]. Similarly, the density f = u 2 yy + u 2 x x u yy + u x x u 2 xy + governs integrable Newtonian equations possessing a fifth-order polynomial integral. In Sect. 2, we investigate the integrability aspects of 2D Lagrangians (1). Our main results can be summarised as follows.
• The Euler-Lagrange equation coming from Lagrangian (1) can be represented as a four-component Hamiltonian system of hydrodynamic type (Sect. 2.1). The requirement of its hydrodynamic integrability (which is equivalent to the vanishing of the corresponding Haantjes tensor) leads to an involutive system of third-order PDEs for the Lagrangian density f (Sect. 2.2). Analysis of the integrability conditions reveals that integrable Lagrangians (1) locally depend on six arbitrary functions of one variable. Furthermore, the integrability conditions are themselves integrable-a standard phenomenon in the theory of integrable systems.
here, W is a partial Legendre transform of the Lagrangian density f (Sect. 2.6). This correspondence works both ways: using known solutions of WDVV equations, one can construct new integrable Lagrangians (1). Conversely, integrable Lagrangian densities f give rise to WDVV prepotentials. Examples of this kind are given in Sects. 2.7.5 and 2.7.6 . In Sect. 3, we investigate 3D second-order Lagrangians of the form f (u x x , u xy , u xt , u yy , u yt , u tt ) dxdydt.
Our results can be summarised as follows: • Integrable Lagrangians in 3D are governed by a third-order PDE system for the Lagrangian density f which comes from the requirement that all travelling wave reductions of a 3D Lagrangian to 2D are integrable in the sense of Sect. 2 (Sect. 3.1). • The class of integrable Lagrangians (5) is invariant under the symplectic group Sp (6, R); the Lagrangian density f transforms as a genus three Siegel modular form of weight −1 (Sect. 3.2).
• Examples of integrable Lagrangians (5) are constructed in Sect. 3.3. These include the densities f = u 2 yy − u x x u xt + u 2 x x u yy + u x x u 2 xy + coming from the theory of the dispersionless KP hierarchy (Sect. 3.3).
In Sect. 4, we discuss examples of integrable dispersive deformations of integrable Lagrangian densities (5). The general problem of constructing such deformations is largely open.
Our interest in second-order integrable Lagrangians stems from the study of 3D first-order Lagrangians [23]. It was pointed out in [24] that the generic integrable Lagrangian density f (u x , u y , u t ) is an automorphic function of its arguments. Let us emphasise an important difference between first-order and second-order Lagrangians in 2D: all first-order Lagrangian densities f (u x , u y ) lead to linearisable Euler-Lagrange equations which, therefore, are automatically integrable. On the contrary, for second-order Lagrangian densities f (u x x , u xy , u yy ), the integrability conditions of the corresponding Euler-Lagrange equations are already highly non-trivial.

Integrable Lagrangians in 2D
In this section, we consider second-order integrable Lagrangians of type (1), f (u x x , u xy , u yy ) dxdy.

Hydrodynamic form of Euler-Lagrange equations
The Euler-Lagrange equation (2) is a fourth-order PDE for u(x, y). Setting a = u x x , b = u xy , c = u yy , we can rewrite (2) Introducing the auxiliary variable p via the relations we can rewrite (6) as a first-order four-component conservative system (7) or, in matrix form, Assuming f cc = 0, we obtain a four-component system of hydrodynamic type, Note that the restriction f cc = 0 is only needed for the matrix R to be invertible and is not really important. One can always make a suitable linear change of the independent variables x, y to transform the Euler-Lagrange equation into a hydrodynamic form. (7) can be put into a Hamiltonian form. For that purpose, we introduce the new dependent variables (A, B, C) which are related to (a, b, c) via partial Legendre transform,

Remark 1 System
In the new variables, system (7) takes the form (P = p) which is manifestly Hamiltonian: with the Hamiltonian density H = h(A, B, C) + B P.

Integrability conditions
Since hydrodynamic-type system (7) is conservative, its integrability by the generalised hodograph method [37,38] is equivalent to the diagonalisability of the matrix V(w) from (8). This is equivalent to the vanishing of the corresponding Haantjes tensor [25]. Recall that the Nijenhuis tensor of the matrix V(w) = (v i j (w)) is defined as where we adopt the notation w = (w 1 , w 2 , w 3 , w 4 ) T = (a, b, c, p) T . The Haantjes tensor is defined as It is easy to see that both tensors are skew-symmetric in the low indices. The requirement of vanishing of the Haantjes tensor leads to a system of PDEs (integrability conditions) for the Lagrangian density f (a, b, c) which can be represented in a symmetric conservative form:  which satisfy the obvious consistency conditions such as Taking these consistency conditions along with the integrability conditions (10), also rewritten in terms of s -variables, we obtain a system of twelve first-order quasilinear equations for s i (a, b, c) which can be represented in the form of two six-component systems of hydrodynamic type, where P, Q are the following 6 × 6 matrices: Direct calculation shows that systems (11) commute, that is, s ab = s ba . Thus, Eq. (11) are in involution, and their general common solution depends on six arbitrary functions of one variable, namely the Cauchy data s i (0, 0, c). This finishes the proof. (10) and (12) imply that there exists a potential ρ such that

Remark 2 Relations
Remark 3 System (10) possesses a Lax pair where λ is a spectral parameter and the 4 × 4 matrices K, L, M are defined as here the potential ρ is defined by relations (13).

Remark 4
We have verified that both commuting six-component systems (11) satisfy the following properties: • They are linearly degenerate, that is, Lie derivatives of the eigenvalues of matrices P, Q along the corresponding eigenvectors are zero, see, for example, [29]. The condition of linear degeneracy can be efficiently verified using a criterion from [19]. • They are non-diagonalisable (their Haantjes tensor does not vanish).
This suggests that integrable Lagrangian densities (1) are related to the associativity (WDVV) equations where analogous systems were obtained in [22], see also [33,34] for related results. Such a link indeed exists and is discussed in Sect. 2.6.

Equivalence group in 2D
Let U be the 2 × 2 Hessian matrix of the function u(x, y). Integrable Lagrangians of type (1) are invariant under Sp(4, R)-symmetry where the matrix

A B C D
belongs to the symplectic group Sp(4, R) (here A, B, C, D are 2×2 matrices). Symmetry (14) 1 originates from linear symplectic transformations of the four-dimensional space with coordinates u x , u y , x, y, which all come from contact (Legendre-type) transformations of the first-order jet space J 1 (x, y, u, u x , u y ), see, for example, [21] where analogous transformations of the Hessian matrix U appeared in the context of integrable Hirota-type equations. Writing the corresponding linear symplectic transformation explicitly in the form and differentiating the second set of equations, we obtain dx which explains the transformation law (14) . Similar equivalence transformations appeared in the context of first-order Lagrangians discussed in [23,24]. Symmetry (14) suggests a relation to Siegel modular forms (the density f transforms as a genus two Siegel modular form of weight −1). Furthermore, integrable Lagrangians (1) are invariant under rescalings of f , as well as under the addition of a 'null Lagrangian', namely transformations of the form which do not affect the Euler-Lagrange equations (here λ i are arbitrary constants).
Transformations (14) and (15) generate a group of dimension 10 + 6 = 16 which preserves the class of integrable Lagrangians (1); this group act as a point symmetry group of integrability conditions (10). Equivalence transformations will be utilised to simplify the classification results in Sect. 2.7, for instance, modulo equivalence transformations, the Lagrangian density (3) is related to f = u xy (u 2 x x − u 2 yy ).
Here, we mainly follow [26,30], which specialised the general results of [17] to the genus two case. Let us introduce two matrix differential operators (here a, b, c and their tilded versionsã,b,c are the six independent variables), and define the operators P 0 , P 1 , P 2 via the expansion Explicitly, we have Let us also define two operators Y 1 , Y 2 depending on the auxiliary parameters ξ = (ξ 1 , ξ 2 ) by the formulae Then integrability conditions (10) can be represented in the Hirota-type bilinear form Here, the left-hand side is a homogeneous quartic in ξ 1 , ξ 2 , with five non-trivial components. Equating them to zero, we obtain all of the five integrability conditions (10). (14), that is, as a weight −1 Siegel modular form, then the left-hand side of (16) transforms as a vector-valued Siegel modular form with values in the representation Sym 4 ⊗ det of GL(2, C).

Remark 6
The principal symbol of the Euler-Lagrange Eq. (2) is given by a compact formula in terms of the operator Y 1 : This expression transforms as a vector-valued Siegel modular form with values in the representation Sym 4 ⊗ det −1 .

Integrable Lagrangians and classical Newtonian equations
Here, we sketch a derivation of the Lagrangian density (4). Consider a classical Newtonian equationẍ , and dot denotes differentiation by t. This equation can be written in the canonical Hamiltonian forṁ To be Liouville integrable, this Hamiltonian system should be equipped with a first integral F(t, x, p) such that First integrals F polynomial in the momentum p were investigated in [10] and later in [28,32]. In particular, the following cases have been studied: In the last (fifth-order) case, Eq. (17) implies the following quasilinear system for the coefficients: Let us introduce a potential u such that U = u x x , V = −u xt , W = 3 2 u 2 x x + u tt . Then the first two equations will be satisfied identically, while the last two imply The compatibility condition of these equations for Q leads to a fourth-order PDE for u, which is nothing but the Euler-Lagrange equation for the second-order Lagrangian whose density is identical to (4) up to relabelling t ↔ y.

Integrable Lagrangians and WDVV equations
Let F(t 1 , . . . , t n ) be a function of n independent variables such that the symmetric matrix is constant and non-degenerate (thus, t 1 is a marked variable), and the coefficients satisfy the associativity condition c s i j c r sk = c s k j c r si ; here, i, j, k ∈ {1, . . . , n}. These requirements impose a nonlinear system of third-order PDEs for the prepotential F, the so-called associativity (WDVV) equations which were discovered in the beginning of 1990s by Witten, Dijkgraaf, Verlinde and Verlinde in the context of two-dimensional topological field theory [9,39]. Geometry and integrability of WDVV equations has been thoroughly studied by Dubrovin, culminating in the remarkable theory of Frobenius manifolds [13]. An important ingredient of this theory is an integrable hydrodynamic hierarchy whose 'primary' part is defined by n commuting Hamiltonian flows where T α are the higher 'times'; note that T 1 = X so that T 1 -flow is trivial: The flows (18) are manifestly Hamiltonian with the Hamiltonian operator η is d dX and the Hamiltonian density ∂ α F. The WDVV equations are equivalent to the requirement of commutativity of these flows.
We will need a particular case of the general construction when n = 4 and the matrix η is anti-diagonal, which corresponds to prepotentials The corresponding primary flows (18) take the form which are Hamiltonian systems with the Hamiltonian densities respectively. In compact form, Eq. (20) can be represented as Setting (t 1 , t 2 , t 3 , t 4 ) = (P, B, C, A), we obtain W (B, C, A).
In this case, WDVV equations reduce to the following system of four PDEs for W : The corresponding primary flows (20) take the form Note that system (22) 2 coincides with (9) under the identification h = W C , thus establishing a link between WDVV equations and integrable Lagrangians. This link can be summarised as follows: • Take prepotential of type (19), set (t 2 , t 3 , t 4 Examples of calculations of this kind will be given in Sect. 2.7.5. f (a, b, c), the corresponding prepotential W (A, B, C) can be reconstructed in quadratures via the formulae

Examples of integrable Lagrangians in 2D
In this section, we present explicit examples of integrable Lagrangian densities f obtained by assuming a suitable ansatz for f and computing the corresponding integrability conditions (10). This gives a whole range of integrable densities expressible via polynomials, elementary functions, Jacobi theta functions and dilogarithms.

Integrable Lagrangian densities of the form f = g(u xx , u yy )
In this case, the integrability conditions lead to the only constraint g aa g cc − g 2 ac = k where k = const. Its solutions can be represented parametrically; thus, for k = 0 (parabolic case) and k = −1 (hyperbolic case) we obtain the general solution in parametric form: respectively; here, p and q are arbitrary functions, and prime denotes differentiation.

Lagrangian densities for linearisable Euler-Lagrange equations
Applying transformations from the equivalence group to quadratic densities f (which generate linear Euler-Lagrange equations), we obtain Lagrangian densities whose Euler-Lagrange equations are linearisable. To state their explicit form, we introduce the variables (r 0 , r 1 , r 2 , r 3 , r 4 ) = (1, a, b, c, ac − b 2 ), the minors of the Hessian matrix U. These variables satisfy the quadratic relation r 0 r 4 − r 1 r 3 + (r 2 ) 2 = 0 which defines the Plücker image of the Lagrangian Grassmannian . Let P i j be the 5 × 5 symmetric matrix of this quadratic form, and let P i j be its inverse: Lagrangian densities with linearisable Euler-Lagrange equations are given by the formula where the quadratic form Q and the covector S satisfy the following conditions: • Covector S is null with respect to the quadratic form P, that is, P i j S i S j = 0. Explicitly, 4S 0 S 4 − 4S 1 S 3 + S 2 2 = 0. (Equivalently, S belongs to the dual variety * .) • The dual vector S i = P i j S j lies in the kernel of the quadratic form Q, that is, Q i j S j = 0. Equivalently, the space dual to T S * (tangent space to * at S) lies in the kernel of Q.
Note that there also exist integrable densities of type (23) which do not satisfy the above conditions.

Integrable Lagrangian densities of the form f = e u xx g(u xy , u yy )
We will show that the generic integrable density of this form corresponds to where is the modular discriminant and θ 1 is the Jacobi theta function. The details are as follows. Substituting f = e a g(b, c) into the integrability conditions (10), one obtains gg ccc = g c g cc + 2g cc g bb − 2(g bc ) 2 , This over-determined system for g is in involution and can be solved as follows. First of all, equation (25) implies so that one can set where h is a function of c only. Using (28), both (24) and (27) reduce to here, prime denotes differentiation by c. Modulo (28) and (29), equation (26) implies Note that (30) can be obtained from (29) by differentiating it with respect to c and using (28), (29). Similarly, differentiating (30) with respect to c we obtain the Chazy equation [6] for h: Equations (29) and (30) can be simplified by the substitution v = −(ln g) bb , which implies v bb = 6v 2 + 4hv + 4h (32) and respectively. Since (32) follows from (33) via differentiating with respect to b, we end up with the following compact form of integrability conditions (24)- (27): here h solves the Chazy equation (31). We recall that modulo the natural SL(2, R)symmetry [7], the Chazy equation possesses three non-equivalent solutions: h = 0, h = 1 and h = 1 2 where is the modular discriminant. These three solutions (which correspond to rational, trigonometric and elliptic cases of the Weierstrass ℘function equation in (34)) are considered separately below. Note that both the rational and trigonometric cases lead to degenerate Lagrangians, so only the elliptic case is of interest.
Rational case h = 0. In this case, Eq. (34) simplify to which are straightforward to solve. Modulo unessential constants the generic solution of these equations is g = e 2μb+μ 2 c (b + μc) where μ = const. The corresponding Lagrangian density f takes the form f = e u xx +2μu xy +μ 2 u yy (u xy + μu yy ).
Note that the change of independent variables x =x, y =ỹ + μx brings this Lagrangian to the degenerate formf = e uxx uxỹ. (The order of the corresponding Euler-Lagrange equation can be reduced by two by setting v = ux .) Trigonometric case h = 1. In this case, Eq. (34) simplify to which are also straightforward to solve. Modulo unessential constants the generic solution of these equations is g = e 2μb+μ 2 c sinh(b + μc) where μ = const. The corresponding Lagrangian density f takes the form f = e u xx +2μu xy +μ 2 u yy sinh(u xy + μu yy ).
Note that the same change of variables as in the rational case brings this Lagrangian to the degenerate formf = e uxx sinh uxỹ.
Elliptic case h = 1 2 , see, for example, [36]. Here the modular discriminant is given by the formula recall that h has the q-expansion 8 w(b, c), we see that the first Eq. (34) becomes the heat equation for w: The general solution of system (35) was constructed in [5]: where σ is the Weierstrass sigma function with the invariants

Remark 8
An alternative (real-valued) representation of the general solution of system (35) in terms of the Jacobi theta function θ 1 is as follows: Note that the function −1/8 (τ )θ 1 (z, τ ) appears in the theory of weak Jacobi forms.
(It is a holomorphic weak Jacobi form of weight −1 and index 1/2.) We refer to [18] for the theory of (weak) Jacobi forms.

Integrable Lagrangian densities polynomial in e u xx and e u yy
Here, we describe integrable Lagrangian densities f that are linear/quadratic in e u xx and e u yy , the coefficients being functions of u xy only.
Linear case: Substituting this ansatz into the integrability conditions (and assuming p 1 , p 2 to be nonzero), we obtain a system of ODEs for the coefficients p i (b) which, modulo equivalence transformations, can be simplified to here α = const (which can be set equal to 1 if nonzero) and prime denotes differentiation by b. Modulo equivalence transformations, these equations possess two essentially different solutions: where the function q(b) satisfies q = 1 sinh b . This implies q = ln 1−e b 1+e b , and another integration gives where Li 2 is the dilogarithm function: (Li 2 (x)) = − ln(1−x) x .
Quadratic case: f = p 0 + p 1 e a + p 2 e c + p 3 e 2a + p 4 e a+c + p 5 e 2c .
Substituting this ansatz into the integrability conditions we obtain a large system of ODEs for the coefficients p i (b) which, modulo equivalence transformations, leads to the following integrable densities (here we only present those examples that are not reducible to the linear case by a change of variables):

Integrable Lagrangian densities from WDVV prepotentials
In this section, we discuss polynomial prepotentials F of type (19) associated with finite Coxeter groups W as given in [14], p. 107. Applying the procedure outlined at the end of Sect. 2.6, we compute the corresponding integrable Lagrangian densities f which, in general, will be algebraic functions of a, b, c (presented below up to appropriate scaling factors).
Group W (A 4 ): Swapping t 2 and t 3 (which is an obvious symmetry of WDVV equations) and following the same procedure gives a polynomial density f : Group W (B 4 ): ; where C is defined by the quadratic equation 3aC 2 +(6a 3 +2b)C + 6 5 a 2 (6a 3 +5b) = c. Swapping t 2 and t 3 gives a polynomial density f : Group W (D 4 ): Group W (F 4 ): Swapping t 2 and t 3 gives a rational density f : Group W (H 4 ): where C is defined by the cubic equation a 12 C 3 + a 7 90 C 2 + a 3 b 3 C + a 13 2·3 5 ·5 2 C + a 9 b 3 4 ·5 + 4 a 19 3 8 ·5 3 ·19 = c. Swapping t 2 and t 3 gives a rational density f : Non-polynomial prepotentials (19) associated with extended affine Weyl groups can be found in [11]: Swapping t 2 and t 3 gives: Modular prepotentials [2,31] give rise to modular Lagrangian densities (as an example we took prepotential 4.2.2. from [31]): Swapping t 2 and t 3 gives: where C is defined by the algebraic equation Here where the functions of γ (a) and g 1 (a) satisfy the ODEs γ = 1 2 γ 2 − 72K g 4 1 , g 1 = 2γ g 1 − g 1 γ , K = const. The above ODE system falls within Bureau's class and its solutions are given in terms of the Schwarzian triangle functions [31].

WDVV prepotentials from integrable Lagrangian densities
In view of the correspondence between integrable Lagrangians and WDVV prepotentials described in Sect. 2.6, integrable Lagrangian densities f (a, b, c) constructed in this paper give rise to prepotentials some of which are apparently new. Here we list some examples (omitting details of calculations; we will only present the corresponding function W ).

Example 1
The polynomial Lagrangian density from Sect. 2, gives rise to the prepotential gives rise to the prepotential the density gives rise to the prepotential Here where Li 2 is the dilogarithm function.

Integrable Lagrangians in 3D
In this section, we consider second-order integrable Lagrangians of the form (5),

Integrability conditions
Let us require that all travelling wave reductions of a 3D Lagrangian density to two dimensions are integrable in the sense of Sects. 2.2 and 2.4 . This gives the necessary conditions for integrability which, in our particular case, prove to be also sufficient. The computational details are as follows. Consider a travelling wave reduction of a 3D Lagrangian density f (u 11 , u 12 , u 22 , u 13 , u 23 , u 33 ) obtained by setting u(x 1 , x 2 , x 3 ) = v(x, y)+ Q where x = s 1 x 1 +s 3 x 3 , y = s 2 x 2 +s 3 x 3 , s i = const, and Q is an arbitrary homogeneous quadratic form in x 1 , x 2 , x 3 . We have where ζ i are the coefficients of the quadratic form Q.
Setting v x x = a, v xy = b, v yy = c, we obtain the reduced 2D Lagrangian density f in the form We have the following differentiation rules: etc. Substituting partial derivatives of the reduced density f (a, b, c) into the 2D integrability conditions (10), we obtain homogeneous polynomials of degree ten in s 1 , s 2 , s 3 whose coefficients are expressed in terms of partial derivatives of the original 3D density f (u i j ). Equating to zero the coefficients of these polynomials, we obtain 3D integrability conditions for f . (Note that due to the presence of arbitrary constants ζ i the arguments of f can be viewed as independent of s 1 , s 2 , s 3 .) The integrability conditions can be represented in compact Hirota-type form analogous to (16): Here the operators on the left-hand side of (37) etc. The left-hand side of (37) is an Sp(6)-invariant operation which transforms a function f defined on the space of 3 × 3 symmetric matrices u i j into a homogeneous form of degree four in ξ 1 , ξ 2 and degree ten in s 1 , s 2 , s 3 .

Equivalence group in 3D
Let U be the 3×3 Hessian matrix of the function u(x 1 , x 2 , x 3 ). Integrable Lagrangians of type (5) are invariant under Sp(6)-symmetry where U σ denote all possible minors of the Hessian matrix U. Transformations (38) and (39)

Dispersive deformations of integrable Lagrangian densities
Some integrable Lagrangian densities f possess integrable dispersive deformations f of the form where f is a second-order integrable density, is a deformation parameter and f i are higher-order differential expressions in u of total degree i (we say that second-order derivatives of u have degree zero, third-order derivatives of u have degree 1, etc). We require that the Euler-Lagrange equation of the deformed density f is integrable (in the usual solitonic sense), so that f can be seen as a dispersionless limit of f when → 0. Dispersive deformations regularise the gradient catastrophe that typically occurs for dispersionless equations, and have been extensively studied by Dubrovin and his School in 1 + 1 dimensions, see, for example, [12] and references therein.
A complete classification of integrable dispersive deformations of second-order Lagrangians is a challenging open problem. The three examples below come from higher-order flows of the KP hierarchy (and provide compact Lagrangian representations thereof). The corresponding dispersive deformations are therefore exact.

Concluding remarks
Here we list some problems for further study.
• Multi-dimensional Lagrangians. It would be of interest to describe multidimensional versions of second-order integrable Lagrangians. Thus, anti-self-dual four-manifolds with a parallel real spinor are described by the integrable 4D Dunajski system [16] a xt + a yz + u x x a yy + u yy a x x − 2u xy a xy = 0, u xt + u yz + u yy u x x − u 2 xy = a, which can be written as a single fourth-order PDE for the function u. This PDE comes from the second-order Lagrangian (u xt + u yz + u yy u x x − u 2 xy ) 2 dxdydzdt.
Similarly, anti-self-dual scalar-flat four-manifolds (Flaherty-Park spaces, see [35] and references therein) are governed by the equations u xz (ln ) yt − u xy (ln ) zt − u zt (ln ) xy + u yt (ln ) xz = 0, u xz u yt − u xy u zt = , which are equivalent to a single fourth-order PDE for u. The corresponding Lagrangian is where one has to substitute = u xz u yt − u xy u zt .
• Multi-component Lagrangians. Our approach can be generalised in a straightforward way to describe 2-field integrable Lagrangians of the form f (u x , u y , v x , v y ) dxdy, as well as their 3D analogues, • Higher-order quasilinear PDEs. Similarly, one can classify third-order integrable PDEs of the form a 1 u x x x + a 2 u xxy + a 3 u xyy + a 4 u yyy = 0 where the coefficients a i are functions of the second-order derivatives u x x , u xy , u yy only. This problem also has a natural 3D analogue.