Deformations of dispersionless Lax systems^*

Let

$$\begin{align*} \tau\colon U\to \mathcal{M} \end{align*}$$

be a vector bundle over a manifold $\mathcal{M}$ and let

$$\begin{align*} \pi_\mathcal{M}\colon \mathcal{B}\to \mathcal{M} \end{align*}$$

be a rank-1 fiber bundle over $\mathcal{M}$. Denote by

$$\begin{align*} \hat\tau\colon\hat U\to\mathcal{B} \end{align*}$$

the pullback bundle of U. Then, for any section u of U, there is a corresponding pullback section $\hat u$ of $\hat U$. Moreover, let $L_0,\ldots, L_s$ be differential operators acting on sections of $\hat U$ with values in the tangent bundle $T\mathcal{B}$,

$$\begin{align*} L_i\in \mathrm{Diff}\left(\hat U,T\mathcal{B}\right). \end{align*}$$

Hence, for a given a section u of U, $L_0(\hat u),\ldots, L_s(\hat u)$ are vector fields on $\mathcal{B}$ depending on u and its derivatives up to certain order.

Definition. We say that u is a solution to the Lax system defined by $(L_i)_{i = 0,\ldots,s}$ if the distribution

$$\begin{align*} D_u = \operatorname{span}\left\{L_0\left(\hat u\right),\ldots,L_s\left(\hat u\right)\right\} \end{align*}$$

on $\mathcal{B}$ is integrable. The solution space of the Lax system will be denoted $\mathfrak{U}_{(L_0,\ldots,L_s)}$.

The fibers of $\mathcal{B}$ typically come with a given parametrization denoted as λ, which is often referred to as the spectral parameter. Later, for simplicity, we shall write $L_i(u)$ instead of $L_i(\hat u)$. The Lax systems are usually defined by pairs $(L_0,L_1)$ of vector fields. The foliation of integral leaves of D_u will be denoted $\mathcal{F}_u$. We stress that $\mathcal{F}_u$ essentially depends on a solution $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$.

For the rest of the paper we shall assume that U and $\mathcal{B}$ are trivial bundles

$$\begin{align*} U = \mathcal{M}\times \mathbb{R}^m\qquad \mathrm{and}\qquad \mathcal{B} = \mathcal{M}\times\mathbb{R} P^1. \end{align*}$$

Moreover, we shall study $\mathcal{M}$ locally around a given point and hence we will assume for simplicity that $\mathcal{M} = \mathbb{R}^n$. The later identification gives the initial coordinates on $\mathcal{M}$.

Let $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$ be a solution to a Lax system $(L_i)_{i = 0,\ldots,s}$ on $\mathcal{M}$. The corresponding twistor space is defined as the quotient space $\mathcal{T}_u = \mathcal{B}/\mathcal{F}_u$ of the foliation $\mathcal{F}_u$ associated to u. Let

$$\begin{align*} \pi_{\mathcal{T}_u}\colon \mathcal{B}\to \mathcal{T}_u, \end{align*}$$

be the quotient mapping.

We assume that fibers of $\pi_\mathcal{M}$ intersect fibers of $\pi_{\mathcal{T}_u}$ transversally, i.e. all $L_i(u)$ for $i = 0,\ldots,s$, are nowhere tangent to the fibers of $\pi_\mathcal{M}$. It follows that one can establish a correspondence between points in $\mathcal{M}$ and certain curves in $\mathcal{T}_u$, and conversely a correspondence between points in $\mathcal{T}_u$ and submanifolds in $\mathcal{M}$. Indeed, let $x\in \mathcal{M}$ and denote by $\hat\gamma_x$ the curve in $\mathcal{B}$ defined as the fiber $\pi^{-1}_\mathcal{M}(x)$. Let $\gamma_{x,u} = \pi_{\mathcal{T}_u}(\hat\gamma_x)$. Then $\gamma_{x,u}$ is a curve in $\mathcal{T}_u$. Conversely, let $p\in \mathcal{T}_u$ and denote by $\hat F_p$ the submanifold in $\mathcal{B}$ defined as the fiber $\pi^{-1}_{\mathcal{T}_u}(p)$. Let $F_p = \pi_\mathcal{M}(\hat F_p)$. Then F_p is a submanifold of $\mathcal{M}$. For a given solution u, we denote

$$\begin{align*} \Gamma_u = \left\{\gamma_{x,u}\ |\ x\in\mathcal{M}\right\}. \end{align*}$$

Our aim now is to introduce a class of distinguished coordinates on $\mathcal{M}$ parameterized by solutions u of a Lax system. It is sufficient to find a map from $\Gamma_u$ to $\mathbb{R}^n$ for any solution $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$, because of the correspondence between points in $\mathcal{M}$ and $\Gamma_u$.

Definition. A collection of corank-1 submanifolds $(T_1, T_2,\ldots,T_m)$ of $\mathcal{T}_u$ is called a transversal system for a family $\Gamma_u$ if any $\gamma\in\Gamma_u$ and T_i , $i = 1,\ldots,m$, intersect transversally exactly at one point.

Definition. Let $T_u = (T_1,\ldots,T_m)$ be a transversal system of submanifolds of $\mathcal{T}_u$ for a family $\Gamma_u$ and assume that any T_l possess a fixed local coordinate system $(x_l^1,\ldots,x_l^k)\colon T_l\to\mathbb{R}^k$. Any function $x_l^j$ defines a function on $\Gamma_u$ by the formula

$$\begin{align*} x_l^j\left(\gamma\right) = x_l^j\left(\gamma\cap T_l\right),\qquad\mathrm{for}\quad \gamma\in\Gamma_u, \end{align*}$$

where the right hand side is the value of $x_l^j$ at the intersection point of γ and T_l (unique by definition), see figure 1. We say that maps $(x_l^j)$ induce coordinates on $\Gamma_u$ if there is a subset $\{x^1,\ldots,x^n\}\subset\{x_l^j\ |\ l = 1,\ldots,m,\ j = 1,\ldots,k\}$ defining local coordinates on $\Gamma_u$. If this is the case then $\phi_u = (x^1,\ldots,x^n)\colon \Gamma_u\to \mathbb{R}^n$ are called induced coordinates on $\mathcal{M}$.

Zoom In Zoom Out Reset image size

The coordinate functions obtained from the construction are very special. Indeed, we have the following.

Proposition 2.1. If $(x^i)_{i = 1,\ldots,n}$, is a coordinate system on $\mathcal{M}$ induced by a transversal system on T_u , then all foliations $x^i = \mathrm{const}$, $i = 1,\ldots,n$, are tangent to the projections from $\mathcal{B}$ to $\mathcal{M}$ of certain integral leaves of $\mathcal{F}_u$.

Proof. Indeed, a condition $x^i = c$ for some $c\in\mathbb{R}$ fixes a submanifold $S\subset T_s$, where T_s is one of the submanifolds from the transversal system T_u . Points $p\in S$ correspond to submanifolds $F_p\subset\mathcal{M}$ that consist of points represented by curves from $\Gamma_u$ intersecting S. On the other hand these submanifolds are, by definition, projections to $\mathcal{M}$ of integral submanifolds of D_u . □

Let $(L_i)_{i = 0,\ldots,s}$ be a Lax system on $\mathcal{M}$. Our goal is to consider two different transversal systems, $T_u^1$ and $T_u^2$, on the twistor space $\mathcal{T}_u$, for any solution $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$ (see figure 2). The two transversal systems give rise to two induced coordinate systems $\phi_u^1\colon\mathcal{M}\to\mathbb{R}^n$ and $\phi_u^2\colon\mathcal{M}\to\mathbb{R}^n$, respectively.

Zoom In Zoom Out Reset image size

The coordinate systems $\phi_u^1$ and $\phi_u^2$ are parameterized by solutions $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$ and, a priori, take values in different copies of $\mathbb{R}^n$. However, we can as well assume that all maps $\phi^1_u$, $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$, take values in a one fixed copy of $\mathbb{R}^n$, denoted $\mathcal{M}_1$, and similarly all maps $\phi^2_u$, $u\in\mathfrak{U}_{(L_0,\ldots,L_s)}$, take values in another fixed copy of $\mathbb{R}^n$, denoted $\mathcal{M}_2$. If it is the case, we can consider $L^1_{i,u} = \phi^1_{u*}L_i$ and $L^2_{i,u} = \phi^2_{u*}L_i$, defined on $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively, and we look for new Lax systems: $(L^1_0,\ldots,L^1_s)$ on $\mathcal{M}_1$ and $(L^2_0,\ldots,L^2_s)$ on $\mathcal{M}_2$ such that

$$\begin{align*} L^1_{i,u} = L^1_i\left(u\circ\left(\phi^1_u\right)^{-1}\right)\qquad \mathrm{and}\qquad L^2_{i,u} = L^2_i\left(u\circ\left(\phi^2_u\right)^{-1}\right). \end{align*}$$

The two Lax system, $(L^1_0,\ldots,L^1_s)$ on $\mathcal{M}_1$ and $(L^2_0,\ldots,L^2_s)$ on $\mathcal{M}_2$, obtained in this manner will be referred to as deformations of the original system. In examples, the procedure follows these steps:

• Step 1.  
  Consider a Lax system in a specific coordinate system and prove that it can be interpreted as an induced coordinate system defined by a transversal system for each solution function u;
• Step 2.  
  Deform all transversal systems and find new coordinates for each solution function u;
• Step 3.  
  Derive a new Lax system based on the formulas in new coordinates.

We introduce the following formal definition.

Definition. Let $(L_i)_{i = 0,\ldots,s}$ be a Lax system on $\mathcal{M}$ and let $\mathfrak{U} = \mathfrak{U}_{(L_0,\ldots,L_s)}$ be its solution space. A Lax system $(\tilde L_i)_{i = 0,\ldots,s}$ on $\tilde{\mathcal{M}}$ is a weak deformation defined by a family of transversal systems $(T_u)_{u\in\mathfrak{U}}$ and induced coordinates $(\phi_u)_{u\in\mathfrak{U}}$ if for any solution $u\in \mathfrak{U}$ the pullback

$$\begin{align*} u^\sharp : = u\circ\phi^{-1}_u \end{align*}$$

to $\tilde{\mathcal{M}}$ via the corresponding φ_u is a solutions to $(\tilde L_i)_{i = 0,\ldots,s}$.

Let us point out that in the definition above there is a different transformation φ_u for each solution function u. On the other hand, we assume that they all map $\mathcal{M}$ to $\tilde{\mathcal{M}}$, which is a fixed manifold, and therefore we can consider a new Lax system on $\tilde{\mathcal{M}}$

Definition. A Lax system $(\tilde L_i)_{i = 0,\ldots,s}$ on $\tilde{\mathcal{M}}$ is a deformation of a Lax system $(L_i)_{i = 0,\ldots,s}$ on $\mathcal{M}$ if the two systems are mutual weak deformations of each other and

$$\begin{align*} \left(u^\sharp\right)^\sharp = u \end{align*}$$

holds.

We get the following tautological result.

Theorem 2.2. If a Lax system $(\tilde L_i)_{i = 0,\ldots,s}$ on $\tilde{\mathcal{M}}$ is a deformation of a Lax system $(L_i)_{i = 0,\ldots,s}$ on $\mathcal{M}$ then there is a one to one correspondence between the two corresponding solution spaces $\mathfrak{U}_{(L_0,\ldots,L_s)}$ and $\mathfrak{U}_{(\tilde L_0,\ldots,\tilde L_s)}$.

Remark. An important aspect of paper [27] is the realizability of vacuum anti-self-dual Einstein metrics as solutions to the equations that arise from the normal forms of Nijenhuis tensors. One can obtain an alternative proof of this result by demonstrating that the equations in [27] are deformations of the heavenly equation (we achieve this in section 3.3), and then apply theorem 2.2. This observation is general: the geometric structures described by deformed Lax systems exhibit the same level of generality as those described by the original ones.

Higher order deformation can be defined by replacing some of T_i in a transversal system with their tangent bundles (or higher order tangent bundles). This can be visualized as a limit, where two (or more) submanifolds T_i in a transversal system become arbitrary close. Then, one can introduce coordinates of a curve γ_x as a first jet (or a higher order jet) at the intersection point of γ_x and T_i . This approach was applied in the case of the Hirota equation in [21]. We shall not explain the details here, but postpone it to separate studies (equations that can be treated in this way include the hyper-CR equation, the Manakov–Santini system and the system that governs the half-integrable Cayley structures introduced in [23]).

It is proved in [8] that the solutions to the dispersionless Hirota equation are in a one to one correspondence with the 3-dimensional hyper-CR Einstein–Weyl structures. The equation, written on manifold $\mathcal{M} = \mathbb{R}^3$ with coordinates $x^1,x^2,x^3$, reads

$$\begin{equation} au_1u_{23}+bu_2u_{13}+cu_3u_{12} = 0, \end{equation} \tag{ 1 }$$

where $a,b,c\in\mathbb{R}$ are non-zero constants such that $a+b+c = 0$, and $u_i = \partial_iu$. The corresponding Lax pair is of the following form (see [8])

$$\begin{equation} L_0 = \partial_3-\frac{u_3}{u_1}\partial_1-\lambda b\partial_3,\qquad L_1 = \partial_2-\frac{u_2}{u_1}\partial_1+\lambda c\partial_2, \end{equation} \tag{ 2 }$$

where λ is an additional affine coordinate on the rank-1 bundle $\mathcal{B} = \mathcal{M}\times\mathbb{R} P^1$. Note that for any value λ, the two vector fields L₀ and L₁ span an integrable distribution on $\mathcal{M}$. Indeed, the Hirota equation is the integrability condition. The corresponding conformal metric $[g]$ and the Weyl connection can be found in [8] (formula (4)).

Let $\lambda_1, \lambda_2,\lambda_3$ be such that $a = \lambda_2-\lambda_3$, $b = \lambda_3-\lambda_1$ and $c = \lambda_1-\lambda_2$. Performing a Möbius transformation $\lambda\mapsto\frac{1}{\lambda_1-\lambda}$ one puts the Lax pair (2) in the form

$$\begin{equation} L_0 = \left(\lambda-\lambda_1\right)\frac{u_3}{u_1}\partial_1-\left(\lambda-\lambda_3\right)\partial_3,\qquad L_1 = \left(\lambda-\lambda_1\right)\frac{u_2}{u_1}\partial_1-\left(\lambda-\lambda_2\right)\partial_2, \end{equation} \tag{ 3 }$$

which clearly gives the same Lax equation for u. One checks that for $\lambda = \lambda_i$ the distribution spanned by L₀ and L₁ is tangent to the foliation $x^i = \mathrm{const}$.

It follows that coordinates $(x^1,x^2,x^3)$ can be interpreted as induced coordinates from a transversal system. Indeed, $\mathcal{T}_u = \mathcal{B}/D_u$ is the space of all integral manifolds of $D_u = \operatorname{span}\{L_0,L_1\}$. Moreover, $\mathcal{T}_u$ is two-dimensional and, as mentioned before, λ is a well defined function on $\mathcal{T}$ because it is constant on leaves of D_u . The transversal system is defined by submanifolds

$$\begin{align*} T_i = \left\{p\in\mathcal{T}_u\ |\ \lambda\left(p\right) = \lambda_i\right\}. \end{align*}$$

Function xⁱ is a diffeomorphism from T_i to $\mathbb{R}$, i.e. it parameterizes leaves of the aforementioned foliation $x^i = \mathrm{const}$ on $\mathcal{M}$.

Let us replace now T_i by general submanifolds of $\mathcal{T}_u$ of the form

$$\begin{align*} \tilde T_i = \left\{p\in\mathcal{T}_u\ |\ f_i\left(p\right) = 0\right\}, \end{align*}$$

for some functions $f_i\colon\mathcal{T}_u\to\mathbb{R}$ on the twistor space (with 0 being a regular value). It is proved in [21] that any choice of coordinates xⁱ on $\tilde T_i$ transforms (1) into

$$\begin{equation} \left(\lambda_2\left(x^2\right)-\lambda_3\left(x^3\right)\right)u_1u_{23}+\left(\lambda_3\left(x^3\right)-\lambda_1\left(x^1\right)\right)u_2u_{13}+\left(\lambda_1\left(x^1\right)-\lambda_2\left(x^2\right)\right)u_3u_{12} = 0, \end{equation} \tag{ 4 }$$

where now each $\lambda_i(x^i)$, $i = 1,2,3$, is a function of one variable. Precisely, λ_i is defined as the restriction of function λ from $\mathcal{T}_u$ to submanifold $\tilde T_i$, and xⁱ is a coordinate function on $\tilde T_i$, which consequently becomes an induced coordinate on $\mathcal{M}$. In the previous case this restriction of λ to T_i is just a constant function. The Lax pair for (4) is given by

$$\begin{align} L_0 = \left(\lambda-\lambda_1\left(x^1\right)\right)\frac{u_3}{u_1}\partial_1-\left(\lambda-\lambda_3\left(x^3\right)\right)\partial_3,\quad L_1 = \left(\lambda-\lambda_1\left(x^1\right)\right)\frac{u_2}{u_1}\partial_1-\left(\lambda-\lambda_2\left(x^2\right)\right)\partial_2. \end{align} \tag{ 5 }$$

Equation (4) was derived for the first time in [17] where the authors analyzed the so-called Nijenhuis operators associated to Veronese webs. The general assumption in their approach is that the Nijenhuis operators have no singular points which is reflected in the condition $\partial_{x^i}\lambda_i(x^i)\neq 0$. In this case a simple coordinate change (a point transformation) gives $\lambda_i(x^i) = x^i$. In our approach $\lambda_i(x^i)$ can be arbitrary smooth function of one variable. In particular we admit $\partial_{x^i}\lambda_i(x^i) = 0$ for certain values of xⁱ .

In [17] it is also proved that equations with constant coefficients λ_i are contactly non-equivalent to the one with non-constant λ_i (one can check that the corresponding symmetry groups are different). Note that the coordinate change between coordinate systems induced by different transversal systems does not establish a contact equivalence of the Lax systems. The reason is that the coordinate change depends on a given solution u (it is a subtle point in the definition of deformations which makes the procedure nontrivial).

On the other hand there is a Bäcklund transformation between solutions to (1) and (4). It was found in [17] and is given by

$$\begin{align*} \lambda_i\tilde\lambda_j\left(x^j\right)u_i\tilde u_j = \lambda_j\tilde\lambda_i\left(x^i\right)u_j\tilde u_i,\quad i,j = 1,2,3, \end{align*}$$

where u and $\tilde u$ are solutions to (1) and (4), respectively, where in (4) $\lambda_i(x^i)$ are replaced by $\tilde\lambda_i(x^i)$ for clarity.

Veronese webs are higher dimensional counterparts of the hyper-CR Einstein–Weyl structures described by equation (1). We refer to [8] for the proof of the 3-dimensional correspondence between the Veronese webs and the Einstein–Weyl structures. The general case was studied in [19] where it is additionally proved that the webs underlie very particular integrable paraconformal structures (so called totally geodesic GL(2)-geometries).

The Veronese webs on $\mathcal{M} = \mathbb{R}^n$ are 1-parameter families of corank-one foliations [14, 18, 31] such that the annihilating 1-forms ω_λ give rise to a rational normal curves $\lambda\mapsto\mathbb{R}\omega_\lambda(x)\in P(T^*_x\mathcal{M})$ at each point $x\in\mathcal{M}$. The curves replace cones of null directions of $[g]$ in the 3-dimensional case. It is proved in [19] that the Veronese webs are in a one to one correspondence with solutions to the following system

$$\begin{equation} \left(\lambda_i-\lambda_j\right)u_ku_{ij}+\left(\lambda_k-\lambda_i\right)u_ju_{jk}+\left(\lambda_j-\lambda_k\right)u_ku_{ij} = 0, \end{equation} \tag{ 6 }$$

where $i,j,k = 1,\ldots,n$. Indeed, the equations are equivalent to the integrability condition

$$\begin{align*} \textrm d\omega_\lambda\wedge\omega_\lambda = 0 \end{align*}$$

where ω_λ is given by

$$\begin{equation} \omega_\lambda = \sum_{i = 1}^n\prod_{j\neq i}\left(\lambda-\lambda_j\right)u_i\textrm dx^i. \end{equation} \tag{ 7 }$$

System (6) appeared in [8] for the first time and turned out to be a Lax system with

$$\begin{equation} L_i = \left(\lambda-\lambda_1\right)\frac{u_i}{u_1}\partial_1-\left(\lambda-\lambda_i\right)\partial_i,\qquad i = 1,\ldots,n. \end{equation} \tag{ 8 }$$

The corresponding twistor space is two dimensional and the most natural transversal system can be defined as before

$$\begin{align*} T_i = \left\{p\in\mathcal{T}\ |\ \lambda\left(p\right) = \lambda_i\right\}, \end{align*}$$

where now $i = 1,\ldots,n$. Note that, as in the 3 dimensional case, λ is constant along vector fields L_i independently of u, and therefore can be treated as a function on the twistor space $\mathcal{T}_u$.

Deforming T_i as in the 3-dimensional case we get that the constants λ_i can be replaced by arbitrary (smooth) functions of one variable $\lambda_i = \lambda_i(x_i)$. Indeed we have

$$\begin{equation} \left(\lambda_i\left(x^i\right)-\lambda_j\left(x^j\right)\right)u_ku_{ij}+\left(\lambda_k\left(x^k\right)-\lambda_i\left(x^i\right)\right)u_ju_{jk}+\left(\lambda_j\left(x^j\right)-\lambda_k\left(x^k\right)\right)u_ku_{ij} = 0, \end{equation} \tag{ 9 }$$

$i,j,k = 1,\ldots,n$, as an integrability condition for ω_λ given by (7) with constants λ_i replaced by functions. Note that, if $\partial_{x^i}\lambda_i\neq 0$ then coordinate change (a point transformation) reduces $\lambda_i(x^i) = x^i$.

Finally, as in the case of the single Hirota equation we find that (6) and (9) are related by a Bäcklund transformation

$$\begin{align*} \lambda_i\tilde\lambda_ju_i\tilde u_j = \lambda_j\tilde\lambda_iu_j\tilde u_i,\qquad i,j = 1,\ldots,n. \end{align*}$$

where $\tilde\lambda_i = \tilde\lambda_i(x^i)$ are functions.

In a recent paper [15] an interesting approach to the heavenly equations is presented. In this context $\mathcal{M} = \mathbb{R}^4$ and one looks for split-signature self-dual Ricci flat metrics. It is well known that metrics of this type are in a one to one correspondence with solutions to the (first) Plebański equation

$$\begin{equation} u_{13}u_{24}-u_{14}u_{23} = 1 \end{equation} \tag{ 10 }$$

which has a Lax pair on $\mathcal{B} = \mathcal{M}\times\mathbb{R} P^1$ of the following form

$$\begin{align*} L_0 = -\partial_3+\lambda\left(u_{13}\partial_2-u_{23}\partial_1\right),\quad L_1 = -\partial_4+\lambda\left(u_{14}\partial_2-u_{24}\partial_1\right). \end{align*}$$

The authors of [15] utilize the eigenfunctions as coordinates, where eigenfunctions are understood as solutions to linear system

$$\begin{equation} L_i|_{\lambda = \lambda^*}\psi = 0, \qquad i = 0,1, \end{equation} \tag{ 11 }$$

for unknown function ψ, where $\lambda^*$ is a fixed value of λ. It follows that any solution ψ is a function on a submanifold of $\mathcal{T}_u$ defined as

$$\begin{align*} T_{\lambda^*} = \left\{p\in\mathcal{T}_u\ |\ \lambda\left(p\right) = \lambda^*\right\}, \end{align*}$$

where as in the previous examples λ is treated here as a function on $\mathcal{T}_u$. Conversely, any function on $T_{\lambda^*}$ is a solution to (11).

In the present case $\mathcal{T}_u$ is 3-dimensional, and consequently $T_{\lambda^*}$ is a 2-dimensional surface in $\mathcal{T}$. It follows that there are two functionally independent functions on each $T_{\lambda^*}$. Konopelchenko et al [15] uses the functions as coordinates on $\mathcal{M}$. In our terminology they are induced coordinates for a transversal system. For the Plebański equation it is enough to take two distinct values of $\lambda^*$, say λ₁ and λ₂, and the corresponding transversal system $T_u = (T_1,T_2)$. We get four functions $x^i_l$, $i,l = 1,2$ in this way, as needed.

However, one can take $T_u = (T_1,T_2,T_3, T_4)$ for four different values of $\lambda^* = \lambda_l$, $l = 1,2,3,4$. Then coordinates on T_l give 8 functions $x^i_l$ in total. We pick 4 of them, one for each λ_l . It is proved in [15] (theorem 4.1) that as a result one gets the general heavenly equation (see also [30])

$$\begin{equation} \left(\lambda_1-\lambda_2\right)\left(\lambda_3-\lambda_4\right)u_{12}u_{34}+\left(\lambda_2-\lambda_3\right)\left(\lambda_1-\lambda_4\right)u_{23}u_{14}+\left(\lambda_3-\lambda_1\right)\left(\lambda_2-\lambda_4\right)u_{31}u_{24} = 0. \end{equation} \tag{ 12 }$$

The corresponding Lax pair has the following form

$$\begin{align*} \begin{aligned} &L_0 = \left(\lambda-\lambda_1\right)\left(\lambda_2-\lambda_3\right)u_{23}\partial_1+\left(\lambda-\lambda_2\right)\left(\lambda_3-\lambda_1\right)u_{13}\partial_2+\left(\lambda-\lambda_3\right)\left(\lambda_1-\lambda_2\right)u_{12}\partial_3,\\ &L_1 = \left(\lambda-\lambda_1\right)\left(\lambda_2-\lambda_4\right)u_{24}\partial_1+\left(\lambda-\lambda_2\right)\left(\lambda_4-\lambda_1\right)u_{14}\partial_2+\left(\lambda-\lambda_4\right)\left(\lambda_1-\lambda_2\right)u_{12}\partial_4. \end{aligned} \end{align*}$$

Now, the transversal system T_u can be deformed to a quadruple $\tilde T_u = (\tilde T_1,\tilde T_2,\tilde T_3, \tilde T_4)$ of arbitrary surfaces (transversal with respect to the family $\Gamma_u$) in $\mathcal{T}_u$. Similarly to the Hirota equation such a deformation replaces each λ_i by a smooth function which in general can be written in the form

$$\begin{equation} \lambda_i = \lambda_i\left(x_i^1,x_i^2\right) \end{equation} \tag{ 13 }$$

where $(x_i^1,x_i^2)$ are two coordinate functions on $\tilde T_i$. One of them, say $x_i^1$, becomes a coordinate on $\mathcal{M}$ (denote it by xⁱ ). Denote by φⁱ the pullback of the second coordinate function (i.e. $x_i^2$) from $\tilde T_i$ to $\mathcal{M}$. Then $\phi^i = \phi^i(x^1,x^2,x^3,x^4)$ is a general function satisfying (11) for $\lambda^* = \lambda_i(x^1_i,x^2_i)$.

The Lax pair in the new coordinate system reads

$$\begin{align*} \begin{aligned} &\tilde L_0 = \left(\lambda-\lambda_1\right)\left(\lambda_2-\lambda_3\right)\tilde U_{23}\partial_1+\left(\lambda-\lambda_2\right)\left(\lambda_3-\lambda_1\right)\tilde U_{13}\partial_2+\left(\lambda-\lambda_3\right)\left(\lambda_1-\lambda_2\right)\tilde U_{12}\partial_3,\\ &\tilde L_1 = \left(\lambda-\lambda_1\right)\left(\lambda_2-\lambda_4\right)\tilde U_{24}\partial_1+\left(\lambda-\lambda_2\right)\left(\lambda_4-\lambda_1\right)\tilde U_{14}\partial_2+\left(\lambda-\lambda_4\right)\left(\lambda_1-\lambda_2\right)\tilde U_{12}\partial_4, \end{aligned} \end{align*}$$

where $\lambda_i = \lambda_i(x_i^1,x_i^2)$ and $\tilde U_{ij}$ are certain expressions involving second order derivatives of the original u and of the corresponding transformation of coordinates. However, it turns out that $\tilde U_{ij}$ can be computed explicitly as

$$\begin{align*} \tilde U_{ij} = \partial_i\partial_j\tilde u \end{align*}$$

where on the right hand side there are second order derivatives with respect to the new coordinates. The formula holds provided that new coordinates, as well as functions φⁱ , are defined as functions on $\tilde T_i$, or equivalently they are solutions to

$$\begin{equation} \tilde L_0|_{\lambda = \lambda_i\left(x^1,\phi^i\right)}\psi = 0, \qquad \tilde L_1|_{\lambda = \lambda_i\left(x^1,\phi^i\right)}\psi = 0, \end{equation} \tag{ 14 }$$

in new coordinates. The integrability condition for the new Lax pair gives (12) (with λ_i upgraded to functions). Notice, that there is one obvious solution to (14) which is $\phi^i = \partial_i\tilde u = \tilde u_i$ (written in new coordinates). Summarizing, if we replace for simplicity $u: = \tilde u$, we get the following equation

$$\begin{align*} &\left(\lambda_1\left(x^1,u_1\right)-\lambda_2\left(x^2,u_2\right)\right)\left(\lambda_3\left(x^3,u_3\right)-\lambda_4\left(x^4,u_4\right)\right)u_{12}u_{34}\\ &\quad +\left(\lambda_2\left(x^2,u_2\right)-\lambda_3\left(x^3,u_3\right)\right)\left(\lambda_1\left(x^1,u_1\right)-\lambda_4\left(x^4,u_4\right)\right)u_{23}u_{14}\\ &\quad +\left(\lambda_3\left(x^3,u_3\right)-\lambda_1\left(x^1,u_1\right)\right)\left(\lambda_2\left(x^2,u_2\right)-\lambda_4\left(x^4,u_4\right)\right)u_{31}u_{24} = 0. \end{align*}$$

In line with the previous examples, one can provide a Bäcklund transformation for the generalized heavenly equation with different values of λ_i , regardless they are constant or not. In the current case, the transformation is expressed via second-order equations:

$$\begin{align*} \left(\lambda_k-\lambda_l\right)\left(\tilde\lambda_i-\tilde\lambda_j\right)u_{kl}\tilde u_{ij} = \left(\tilde\lambda_k-\tilde\lambda_l\right)\left(\lambda_i-\lambda_j\right)\tilde u_{kl}u_{ij}. \end{align*}$$

Note that equations are linear with respect to u, provided that the corresponding λ_i 's are constant.

As mentioned in the Introduction, specific cases of this particular deformation appeared independently in [16, 27]. In [16] authors consider deformations of the symmetry algebra and their algebraic procedure give our deformed equation but only with respect to λ₄ and coordinate x⁴ (it is case (I) of [16]). Function Q in [16] is given by

$$\begin{align*} Q\left(x^4,u_4\right) = \frac{\left(\lambda_1-\lambda_3\right)\left(\lambda_2-\lambda_4\left(x^4,u_4\right)\right)}{\left(\lambda_2-\lambda_3\right)\left(\lambda_1-\lambda_4\left(x^4,u_4\right)\right)}. \end{align*}$$

On the other hand authors of [27] apply approach of the Nijenhuis operators and get $\lambda_i = \lambda_i(x^i)$. This corresponds to the case when the foliation $x^1_i = \mathrm{const}$ on $\tilde T_i$ coincide with the foliation $\lambda = \mathrm{const}$ on $\mathcal{T}_u$ restricted to $\tilde T_i$. As in the previous examples our method covers singularities of the Nijenhuis operators—corresponding to points where $\partial_{x^i}\lambda_i(x^i) = 0$.