# Lax representations via twisted extensions of infinite-dimensional Lie
algebras: some new results

Oleg I. Morozov  Trapeznikov Institute of Control Sciences,
65 Profsoyuznaya Street, Moscow 117997, Russia<EMAIL_ADDRESS>
(Date: April 1, 2022.)

###### Abstract.

We find new integrable partial differential equations with Lax representations
generated by extensions of Lie algebras of the Kac–Moody type as well as the
Lie algebra of Hamiltonian vector fields on $\mathbb{R}^{2}$.

###### Key words and phrases:

twisted cohomology, Maurer–Cartan forms, symmetries of differential equations,
Lax representations

###### 2020 Mathematics Subject Classification:

58H05, 58J70, 35A30, 37K05, 37K10

##  1\. Introduction

Lax representations provide the basic construction that allows applications of
a number of techniques for studying nonlinear partial differential equations
(pdes), whence they are considered as the key feature indicating integrability
thereof, see [36, 37, 34, 29, 12, 1, 20, 32, 5] and references therein.
Therefore the problem of finding intrinsic conditions that ensure existence of
a Lax representation for a given pde is of great importance in the theory of
integrable systems. In the recent papers [22] – [26] we propose an approach to
tackle this problem. We have shown there that for a number of pdes including
the potential Khokhlov–Zabolotskaya equation, the Boyer–Finley equation, the
hyper-CR equation of Einstein–Weyl structures, the reduced quasiclassical
self-dual Yang–Mills equation, the 4D Martínez Alonso–Shabat equation, the 4D
universal hierarchy equation, and other equations, their known Lax
representations can be inferred from non-triviality of the second twisted
cohomology groups of the Lie algebras of contact symmetries of the pdes.
Moreover, we have shown that the technique allows one to find new Lax
representations of some pdes.

The aim of the present paper is to gain a better understanding of the
relationship between the structure theory of infinite-dimensional Lie algebras
and the theory of integrable systems. In particular, we construct new examples
of deriving Lax representations of pdes from twisted extensions of some Lie
algebras.

In a number of above-mentioned examples the symmetry algebras of the pdes have
the form of the semi-direct sum
$\mathfrak{s}_{\diamond}\ltimes\mathfrak{q}_{N,\varepsilon}$ of a finite-
dimensional Lie algebra $\mathfrak{s}_{\diamond}$ and the infinite-dimensional
Lie algebra $\mathfrak{q}_{N,\varepsilon}$ of the Kac–Moody type, [6], that
is, the deformation of the tensor product
$\mathfrak{q}_{N,0}=\mathbb{R}_{N}[s]\otimes\mathfrak{w}$, where
$\mathbb{R}_{N}[s]$ is the commutative associative algebra of truncated
polynomials of degree $N$ and $\mathfrak{w}$ is the Lie algebra of vector
fields on $\mathbb{R}$, see Section 3 for definition of
$\mathfrak{q}_{N,\varepsilon}$. The second twisted cohomology groups of the
Lie algebras $\mathfrak{s}_{\diamond}\ltimes\mathfrak{q}_{N,\varepsilon}$ from
our examples turn out to be nontrivial, and the nontrivial twisted 2-cocycles
generate twisted extensions of these Lie algebras. Linear combinations of the
Maurer–Cartan forms of the twisted extensions provide the Wahlquist–Estabrook
forms that generate Lax representations of the pdes.

In examples from [22]–[26] we have $N\geq 3$, and the natural question is
whether there exist Lie algebras of the form
$\mathfrak{s}_{\diamond}\ltimes\mathfrak{q}_{N,\varepsilon}$ with $N<3$ and
nontrivial second twisted cohomology groups whose twisted extensions generate
Lax representations of some pdes. In Sections 4, 5, and 6 we present three
systems (4.3) - (4.4), (5.5) - (5.6), and (6.2) - (6.3) whose Lax
representations are generated by extensions of the Lie algebras
$\mathfrak{q}_{1,-1}$, $\mathfrak{q}_{1,-2}$, and $\mathfrak{q}_{2,-1}$,
respectively. Equation (5.5) can be considered as a 3D generalization of the
generalized 2D Hunter–Saxton equation [9, 10, 3, 27] with the special value of
the parameter. We compare the structure of the symmetry algebras of the
obtained systems with the structure of the Lie algebras that generate the Lax
representations.

Paper [24] shows that for a number of known 4D integrable equations their Lax
representations can be derived from the twisted extensions of the symmetry
algebras, which turn out to be of the form
$\mathfrak{s}_{\diamond}\ltimes(\mathbb{R}_{N}[s]\otimes\mathbb{R}[t]\otimes\mathfrak{w})$.
In [26] we apply the technique to a Lie algebra of this form and construct a
new 4D integrable equation. In the present paper we address an interesting
question of finding new examples of integrable pdes whose symmetry algebras
include the Lie algebra $\mathfrak{h}$ of Hamiltonian vector fields on
$\mathbb{R}^{2}$ as a subalgebra. In Section 7 we present such an example. We
derive equation (7.4) from Lax representation defined by extensions of the Lie
algebra $\mathfrak{h}\oplus\mathfrak{w}$ . Other examples of integrable
systems whose symmetry algebras include $\mathfrak{h}$ as ‘building blocks’
are given by the family of heavenly equations [33, 4, 7, 18] and their
‘symmetric deformations’ [19]. Hence we refer equation (7.4) to as the
‘degenerate heavenly equation’.

##  2\. Preliminaries and notation

###  2.1. Symmetries and Lax representations

The presentation in this section closely follows [13]—[17] and [35]. All our
considerations are local. Let
$\pi\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$,
$\pi\colon(x^{1},\dots,x^{n},u^{1},\dots,u^{m})\mapsto(x^{1},\dots,x^{n})$, be
a trivial bundle, and $J^{\infty}(\pi)$ be the bundle of its jets of the
infinite order. The local coordinates on $J^{\infty}(\pi)$ are
$(x^{i},u^{\alpha},u^{\alpha}_{I})$, where $I=(i_{1},\dots,i_{n})$ are multi-
indices with $i_{k}\geq 0$, and for every local section
$f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\times\mathbb{R}^{m}$ of $\pi$
the corresponding infinite jet $j_{\infty}(f)$ is a section
$j_{\infty}(f)\colon\mathbb{R}^{n}\rightarrow J^{\infty}(\pi)$ such that
$u^{\alpha}_{I}(j_{\infty}(f))=\displaystyle{\frac{\partial^{\\#I}f^{\alpha}}{\partial
x^{I}}}=\displaystyle{\frac{\partial^{i_{1}+\dots+i_{n}}f^{\alpha}}{(\partial
x^{1})^{i_{1}}\dots(\partial x^{n})^{i_{n}}}}$. We put
$u^{\alpha}=u^{\alpha}_{(0,\dots,0)}$. Also, we will simplify notation in the
following way: e.g., in the case of $n=3$, $m=1$ we denote $x^{1}=t$,
$x^{2}=x$ $x^{3}=y$, and $u^{1}_{(i,j,k)}=u_{{t\dots t}{x\dots x}{y\dots y}}$
with $i$ times $t$, $j$ times $x$, and $k$ times $y$.

The vector fields

$D_{x^{k}}=\frac{\partial}{\partial x^{k}}+\sum\limits_{\\#I\geq
0}\sum\limits_{\alpha=1}^{m}u^{\alpha}_{I+1_{k}}\,\frac{\partial}{\partial
u^{\alpha}_{I}},\qquad k\in\\{1,\dots,n\\},$

$(i_{1},\dots,i_{k},\dots,i_{n})+1_{k}=(i_{1},\dots,i_{k}+1,\dots,i_{n})$, are
called total derivatives. They commute everywhere on $J^{\infty}(\pi)$:
$[D_{x^{i}},D_{x^{j}}]=0$.

The evolutionary vector field associated to an arbitrary vector-valued smooth
function $\varphi\colon J^{\infty}(\pi)\rightarrow\mathbb{R}^{m}$ is the
vector field

$\mathbf{E}_{\varphi}=\sum\limits_{\\#I\geq
0}\sum\limits_{\alpha=1}^{m}D_{I}(\varphi^{\alpha})\,\frac{\partial}{\partial
u^{\alpha}_{I}}$

with $D_{I}=D_{(i_{1},\dots\,i_{n})}=D^{i_{1}}_{x^{1}}\circ\dots\circ
D^{i_{n}}_{x^{n}}$.

A system of pdes $F_{r}(x^{i},u^{\alpha}_{I})=0$ of the order $s\geq 1$ with
$\\#I\leq s$, $r\in\\{1,\dots,R\\}$ for some $R\geq 1$, defines the
submanifold $\mathcal{E}=\\{(x^{i},u^{\alpha}_{I})\in
J^{\infty}(\pi)\,\,|\,\,D_{K}(F_{r}(x^{i},u^{\alpha}_{I}))=0,\,\,\\#K\geq
0\\}$ in $J^{\infty}(\pi)$.

A function $\varphi\colon J^{\infty}(\pi)\rightarrow\mathbb{R}^{m}$ is called
a (generator of an infinitesimal) symmetry of equation $\mathcal{E}$ when
$\mathbf{E}_{\varphi}(F)=0$ on $\mathcal{E}$. The symmetry $\varphi$ is a
solution to the defining system

(2.1) $\ell_{\mathcal{E}}(\varphi)=0,$

where $\ell_{\mathcal{E}}=\ell_{F}|_{\mathcal{E}}$ with the matrix
differential operator

$\ell_{F}=\left(\sum\limits_{\\#I\geq 0}\frac{\partial F_{r}}{\partial
u^{\alpha}_{I}}\,D_{I}\right).$

The symmetry algebra $\mathrm{Sym}(\mathcal{E})$ of equation $\mathcal{E}$ is
the linear space of solutions to (2.1) endowed with the structure of a Lie
algebra over $\mathbb{R}$ by the Jacobi bracket
$\\{\varphi,\psi\\}=\mathbf{E}_{\varphi}(\psi)-\mathbf{E}_{\psi}(\varphi)$.
The algebra of contact symmetries $\mathrm{Sym}_{0}(\mathcal{E})$ is the Lie
subalgebra of $\mathrm{Sym}(\mathcal{E})$ defined as
$\mathrm{Sym}(\mathcal{E})\cap C^{\infty}(J^{1}(\pi))$. Symmetries with
generators of the form
$\varphi^{\alpha}=\eta^{\alpha}-\sum_{i}\xi^{i}\,u^{\alpha}_{i}$,
$\eta^{\alpha},\xi^{i}\in C^{\infty}(J^{0}(\pi))$, are referred to as point
symmetries. They correspond to vector fields
$\sum_{i}\xi^{i}\,\partial_{x^{i}}+\sum_{\alpha}\eta^{\alpha}\,\partial_{u^{\alpha}}$
on $J^{0}(\pi)$.

Let the linear space $\mathcal{W}$ be either $\mathbb{R}^{N}$ for some $N\geq
1$ or $\mathbb{R}^{\infty}$ endowed with local coordinates $w^{a}$,
$a\in\\{1,\dots,N\\}$ or $a\in\mathbb{N}$, respectively. Variables $w^{a}$ are
called pseudopotentials [36]. Locally, a differential covering of
$\mathcal{E}$ is a trivial bundle $\varpi\colon
J^{\infty}(\pi)\times\mathcal{W}\rightarrow J^{\infty}(\pi)$ equipped with
extended total derivatives

$\widetilde{D}_{x^{k}}=D_{x^{k}}+\sum\limits_{a}T^{a}_{k}(x^{i},u^{\alpha}_{I},w^{b})\,\frac{\partial}{\partial
w^{a}}$

such that $[\widetilde{D}_{x^{i}},\widetilde{D}_{x^{j}}]=0$ for all $i\not=j$
whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$. Define the partial
derivatives of $w^{a}$ by $w^{s}_{x^{k}}=\widetilde{D}_{x^{k}}(w^{s})$. This
yields the over-determined system of pdes

(2.2) $w^{a}_{x^{k}}=T^{a}_{k}(x^{i},u^{\alpha}_{I},w^{b})$

which is compatible whenever $(x^{i},u^{\alpha}_{I})\in\mathcal{E}$. System
(2.2) is referred to as the covering equations or the Lax representation of
equation $\mathcal{E}$.

Dually, the differential covering is defined by the Wahlquist–Estabrook forms

(2.3)
$\tau^{a}=dw^{a}-\sum\limits_{k=1}^{m}T^{a}_{k}(x^{i},u^{\alpha}_{I},w^{b})\,dx^{k}$

as follows: when $w^{a}$ and $u^{\alpha}$ are considered to be functions of
$x^{1}$, … , $x^{n}$, forms (2.3) are equal to zero if and only if system
(2.2) holds.

###  2.2. Twisted cohomology of Lie algebras

For a Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, its representation
$\rho\colon\mathfrak{g}\rightarrow\mathrm{End}(V)$, and $k\geq 1$ let
$C^{k}(\mathfrak{g},V)=\mathrm{Hom}(\Lambda^{k}(\mathfrak{g}),V)$ be the space
of all $k$–linear skew-symmetric mappings from $\mathfrak{g}$ to $V$. Then the
Chevalley–Eilenberg differential complex

$V=C^{0}(\mathfrak{g},V)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}C^{1}(\mathfrak{g},V)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\dots\stackrel{{\scriptstyle
d}}{{\longrightarrow}}C^{k}(\mathfrak{g},V)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}C^{k+1}(\mathfrak{g},V)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\dots$

is generated by the differential $d\colon\theta\mapsto d\theta$ such that

$d\theta(X_{1},...,X_{k+1})=\sum\limits_{q=1}^{k+1}(-1)^{q+1}\rho(X_{q})\,(\theta(X_{1},...,\hat{X}_{q},...,X_{k+1}))$
(2.4) $\quad+\sum\limits_{1\leq p<q\leq
k+1}(-1)^{p+q+1}\theta([X_{p},X_{q}],X_{1},...,\hat{X}_{p},...,\hat{X}_{q},...,X_{k+1}).$

The cohomology groups of the complex $(C^{*}(\mathfrak{g},V),d)$ are referred
to as the cohomology groups of the Lie algebra $\mathfrak{g}$ with
coefficients in the representation $\rho$. For the trivial representation
$\rho_{0}\colon\mathfrak{g}\rightarrow\mathbb{R}$, $\rho_{0}\colon X\mapsto
0$, the cohomology groups are denoted by $H^{*}(\mathfrak{g})$.

Consider a Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ with non-trivial first
cohomology group $H^{1}(\mathfrak{g})$ and take 1-form $\alpha\neq 0$ on
$\mathfrak{g}$ such that $d\alpha=0$. Then for each $c\in\mathbb{R}$ define
the twisted differential $d_{c\alpha}\colon
C^{k}(\mathfrak{g},\mathbb{R})\rightarrow C^{k+1}(\mathfrak{g},\mathbb{R})$ by
the formula

$d_{c\alpha}\theta=d\theta-c\,\alpha\wedge\theta.$

From $d\alpha=0$ it follows that $d_{c\alpha}^{2}=0$. The cohomology groups of
the complex

$C^{1}(\mathfrak{g},\mathbb{R})\stackrel{{\scriptstyle
d_{c\alpha}}}{{\longrightarrow}}\dots\stackrel{{\scriptstyle
d_{c\alpha}}}{{\longrightarrow}}C^{k}(\mathfrak{g},\mathbb{R})\stackrel{{\scriptstyle
d_{c\alpha}}}{{\longrightarrow}}C^{k+1}(\mathfrak{g},\mathbb{R})\stackrel{{\scriptstyle
d_{c\alpha}}}{{\longrightarrow}}\dots$

are referred to as the twisted cohomology groups [30, 31] of $\mathfrak{g}$
and denoted by $H^{*}_{c\alpha}(\mathfrak{g})$.

##  3\. Lie algebras of the Kac–Moody type and their extensions

Consider the Lie algebra
$\mathfrak{q}_{N,0}=\mathbb{R}_{N}[s]\otimes\mathfrak{w}$, where
$\mathbb{R}_{N}[s]=\mathbb{R}[s]/\langle s^{N+1}=0\rangle$ is the commutative
unital algebra of truncated polynomials of variable $s$ of degree $N$, and
$\mathfrak{w}=\langle V_{k}\,\,|\,\,k\geq 0\rangle$,
$\displaystyle{V_{k}=\frac{1}{k!}\,t^{k}\,\partial_{t}}$, is the Lie algebra
of polynomial vector fields on $\mathbb{R}$ referred to as the (one-sided)
Witt algebra. Algebra $\mathfrak{q}_{N,0}$ admits the deformation111For the
full description of deformations of the Lie algebra $\mathfrak{q}_{N,0}$ see
[38]. generated by cocycle $\Psi\in
H^{2}(\mathfrak{q}_{N,0},\mathfrak{q}_{N,0})$,

$\Psi(s^{p}\otimes V_{m},s^{q}\otimes
V_{n})=\left\\{\begin{array}[]{lcl}\displaystyle{\frac{p\,n-q\,m}{m+n}}\,{m+n\choose
m}\,s^{p+q}\otimes V_{m+n-1},&&m+n\geq 1,p+q\leq N,\\\
0,&&\mathrm{otherwise}.\end{array}\right.$

For each $\varepsilon\neq 0$ this cocycle defines new bracket
$[\cdot,\cdot]_{\varepsilon}=[\cdot,\cdot]+\varepsilon\,\Psi(\cdot,\cdot)$ on
the linear space $\langle s^{p}\otimes V_{m}\,|\,p\leq N,m\geq 0\rangle$. We
denote the resulting Lie algebra as $\mathfrak{q}_{N,\varepsilon}$. In other
words, the Lie algebra $\mathfrak{q}_{N,\varepsilon}$ is isomorphic to the
linear space of functions $f(t,s)=f_{0}(t)+s\,f_{1}(t)+\dots+s^{N}\,f_{N}(t)$,
$f_{k}\in\mathbb{R}[t]$, equipped with the bracket

(3.1)
$[f,g]_{\varepsilon}=f\,g_{t}-g\,f_{t}+\varepsilon\,s\,(f_{s}\,g_{t}-g_{s}\,f_{t})$

such that there holds $s^{k}=0$ for $k>N$. Likewise to [6] it can be shown
that $\mathfrak{q}_{N,\varepsilon}\subsetneq\mathfrak{g}(A_{M}^{(1)})$ for
some $M\geq N$, see [11] for definition of the Lie algebra
$\mathfrak{g}(A_{M}^{(1)})$. Therefore $\mathfrak{q}_{N,\varepsilon}$ are
referred to as Lie algebras of the Kac–Moody type.

Consider the dual 1-forms $\theta_{p,k}$ to the basis $s^{q}\otimes V_{m}$ of
$\mathfrak{q}_{N,\varepsilon}$, that is, the linear mappings
$\theta_{p,k}\colon\mathfrak{q}_{N,\varepsilon}\rightarrow\mathbb{R}$ such
that $\theta_{p,k}(s^{p}\otimes V_{m})=\delta_{p,q}\,\delta_{k,m}$. Define the
formal series

(3.2)
$\Theta=\sum\limits_{k=0}^{N}\sum\limits_{m=0}^{\infty}\frac{h_{0}^{k}h_{1}^{m}}{m!}\theta_{k,m}$

with formal parameters $h_{0}$ and $h_{1}$ such that $h_{0}^{k}=0$ when $k>N$
and $dh_{0}=dh_{1}=0$. Then (2.4) and (3.1) entail the structure equations

(3.3)
$d\Theta=\Theta_{h_{1}}\wedge(\Theta+\varepsilon\,h_{0}\,\Theta_{h_{0}})$

of the Lie algebra $\mathfrak{q}_{N,\varepsilon}$. Here and below we use
notion $\Theta_{h_{i}}=\partial_{h_{i}}\Theta$ for partial derivatives of the
formal series of 1-forms with respect to the formal parameters $h_{i}$.

For each $N\in\mathbb{N}$ and $\varepsilon\in\mathbb{R}$ the map $D_{0}\colon
s^{p}\otimes v_{k}\mapsto p\,s^{p}\otimes v_{k}$ is an outer derivation of
$\mathfrak{q}_{N,\varepsilon}$. Denote by
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}$ the associated one-
dimensional ‘right’ extension, [8, §1.4.4], of $\mathfrak{q}_{N,\varepsilon}$.
As a vector space $\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}=\langle
Z_{0}\rangle\oplus\mathfrak{q}_{N,\varepsilon}$, and the bracket on
$\mathfrak{q}_{N,\varepsilon}$ is extended to $Z_{0}$ by the formula
$[Z_{0},s^{p}\otimes V_{k}]_{\varepsilon}=D_{0}(s^{p}\otimes
V_{k})=p\,s^{p}\otimes V_{k}$.

Let
$\alpha_{0}\colon\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}\rightarrow\mathbb{R}$
be the dual form to $-Z_{0}$, that is, $\alpha_{0}(Z_{0})=-1$ and
$\alpha_{0}(s^{p}\otimes V_{k})=0$. Then the structure equations for
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}$ read

$\left\\{\begin{array}[]{lcl}d\Theta&=&\Theta_{h_{1}}\wedge(\Theta+\varepsilon\,h_{0}\,\Theta_{h_{0}})+h_{0}\,\alpha_{0}\wedge\Theta_{h_{0}},\\\
d\alpha_{0}&=&0.\end{array}\right.$

For some values of $N$ and $\varepsilon$ the Lie algebras
$\mathfrak{s}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}$ admit further right
extensions. In [22, 23, 24, 25] we have shown examples of integrable pdes
whose Lax representations can be inferred from such extensions when $N\geq 3$.
In the next three sections we consider integrable pdes that are related to
extensions of $\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-1}$,
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-1/2}$, and
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-2}$.

As it was shown in [23], for $N\in\mathbb{N}$ and $\varepsilon=-r^{-1}$ with
$r\in\\{1,\dots,N\\}$ the Lie algebra
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}$ admits the right
extension $\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,\varepsilon}$ generated by
the outer derivation
$D_{1}\colon\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}\rightarrow\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,\varepsilon}$
with

$D_{1}(s^{p}\otimes V_{k})=\left\\{\begin{array}[]{lcl}k\,s^{p+r}\otimes
V_{k-1},&&p+r\leq N,k\geq 1,\\\ 0,&&\mathrm{otherwise}.\end{array}\right.$

We have $[D_{0},D_{1}]=D_{0}\circ D_{1}-D_{1}\circ D_{0}=r\,D_{1}$. Then
$\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,-1/r}$ as a vector space is $\langle
Z_{1}\rangle\oplus(\mathfrak{a}_{0}\ltimes\mathfrak{q}_{N,-1/r})$, with the
extension of the bracket of $\mathfrak{q}_{N,-1/r}$ given by
$[Z_{0},Z_{1}]_{-1/r}=-r\,Z_{1}$ and $[Z_{1},s^{p}\otimes
V_{k}]_{-1/r}=D_{1}(s^{p}\otimes V_{k})$. Consider the dual form $\alpha_{1}$
to the vector $-Z_{1}$, that, put $\alpha_{1}(Z_{1})=-1$,
$\alpha_{1}(Z_{0})=\alpha_{1}(s^{p}\otimes V_{k})=0$. Then the structure
equations for the Lie algebra $\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,-1/r}$
get the form

(3.4)
$\left\\{\begin{array}[]{lcl}d\Theta&=&\displaystyle{\Theta_{h_{1}}\wedge\left(\Theta-\frac{h_{0}}{r}\,\Theta_{h_{0}}-h_{0}^{r}\,\alpha_{1}\right)+h_{0}\,\alpha_{0}\wedge\Theta_{h_{0}},}\\\
d\alpha_{0}&=&\displaystyle{0,\phantom{\frac{A}{A}}}\\\
d\alpha_{1}&=&\displaystyle{r\,\alpha_{0}\wedge\alpha_{1}.\phantom{\frac{A}{A}}}\end{array}\right.$

These equations yield
$H^{1}(\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,-1/r})=\langle\alpha_{0}\rangle$
and $[\alpha_{0}\wedge\alpha_{1}]\in
H^{2}_{r\alpha_{0}}(\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,-1/r})$, therefore
the Lie algebra $\mathfrak{a}_{1}\ltimes\mathfrak{q}_{N,-1/r}$ admits the
twisted extension $\mathfrak{a}_{2}\ltimes\mathfrak{q}_{N,-1/r}$ with the
structure equations obtained by appending equation

(3.5) $d\alpha_{2}=r\,\alpha_{0}\wedge\alpha_{2}+\alpha_{0}\wedge\alpha_{1}$

to system (3.4).

Furthermore, we have $[\alpha_{1}\wedge\alpha_{2}]\in
H^{2}_{2r\alpha_{0}}(\mathfrak{a}_{2}\ltimes\mathfrak{q}_{N,-1/r})$, therefore
the Lie algebra $\mathfrak{a}_{2}\ltimes\mathfrak{q}_{N,-1/r}$ admits the
twisted extension $\mathfrak{a}_{3}\ltimes\mathfrak{q}_{N,-1/r}$ whose
structure equations are obtained by appending equation

(3.6)
$d\alpha_{3}=2\,r\,\alpha_{0}\wedge\alpha_{3}+\alpha_{1}\wedge\alpha_{2}$

to system (3.4), (3.5). This process can be repeated:
$\mathfrak{a}_{k}\ltimes\mathfrak{q}_{N,-1/r}$ admits the twisted extension
$\mathfrak{a}_{k+1}\ltimes\mathfrak{q}_{N,-1/r}$ defined by the twisted
2-cocycle $[\alpha_{1}\wedge\alpha_{k}]\in
H^{2}_{kr\alpha_{0}}(\mathfrak{a}_{k}\ltimes\mathfrak{q}_{N,-1/r})$. In other
words, the extended Lie algebra
$\mathfrak{a}_{k+1}\ltimes\mathfrak{q}_{N,-1/r}$ has the structure equations

(3.7)
$\left\\{\begin{array}[]{lcl}d\Theta&=&\displaystyle{\Theta_{h_{1}}\wedge\left(\Theta-\frac{h_{0}}{r}\,\Theta_{h_{0}}-h_{0}^{r}\,\alpha_{1}\right)+h_{0}\,\alpha_{0}\wedge\Theta_{h_{0}},}\\\
d\alpha_{0}&=&\displaystyle{0,\phantom{\frac{A}{A}}}\\\
d\alpha_{1}&=&\displaystyle{r\,\alpha_{0}\wedge\alpha_{1},\phantom{\frac{A}{A}}}\\\
d\alpha_{m+1}&=&\displaystyle{m\,r\,\alpha_{0}\wedge\alpha_{m+1}+\alpha_{1}\wedge\alpha_{m},\qquad
m\in\\{1,\dots,k\\}.\phantom{\frac{A}{A}}}\end{array}\right.$

##  4\. Integrable equation associated to
$\mathfrak{a}_{2}\ltimes\mathfrak{q}_{1,-1}$

Consider the Lie algebra $\mathfrak{a}_{3}\ltimes\mathfrak{q}_{1,-1}$ defined
by the structure equations (3.7) with $r=1$ and $k=2$, that is, by system

(4.1)
$\left\\{\begin{array}[]{lcl}d\Theta&=&\displaystyle{\Theta_{h_{1}}\wedge\left(\Theta-
h_{0}\,\Theta_{h_{0}}-h_{0}\,\alpha_{1}\right)+h_{0}\,\alpha_{0}\wedge\Theta_{h_{0}},}\\\
d\alpha_{0}&=&\displaystyle{0,\phantom{\frac{A}{A}}}\\\
d\alpha_{1}&=&\displaystyle{\alpha_{0}\wedge\alpha_{1},\phantom{\frac{A}{A}}}\\\
d\alpha_{2}&=&\displaystyle{\alpha_{0}\wedge\alpha_{2}+\alpha_{0}\wedge\alpha_{1},\phantom{\frac{A}{A}}}\\\
d\alpha_{3}&=&\displaystyle{2\,\alpha_{0}\wedge\alpha_{3}+\alpha_{0}\wedge\alpha_{2}.\phantom{\frac{A}{A}}}\end{array}\right.$

Frobenius’ theorem allows one to integrate equations (4.1) step by step. In
particular, we have

$\alpha_{0}=\frac{dq}{q},\quad\alpha_{1}=q\,dy,\quad\alpha_{2}=q\,(dw+\ln
q\,\,dy),\quad\alpha_{3}=q^{2}\,(dv-w\,dy),$
$\theta_{0,0}=a\,dt,\quad\theta_{1,0}=q\,(dx-\ln
a\,\,dy+u\,dt),\quad\theta_{1,1}=q\,a^{-1}\,(du-b\,dy-p\,dt),$

where $q\neq 0$, $a\neq 0$, $t$, $x$, $y$, $u$, $v$, $w$ are free parameters
(‘constants of integration’). We do not need explicit expressions for the
other forms $\theta_{i,j}$ in what follows.

We proceed by imposing the requirement for the linear combination
$\theta_{1,1}-\theta_{1,0}=q\,a^{-1}\,(du-a\,dx-(b-a\,\ln
a)\,dy-(p+a\,u)\,dt)$ to be a multiple of the contact form $du-u_{t}\,dt-
u_{x}\,dx-u_{y}\,dy$ on the bundle of jets of sections of the bundle
$\pi\colon\mathbb{R}^{4}\rightarrow\mathbb{R}^{3}$,
$\pi\colon(t,x,y,u)\mapsto(t,x,y)$, that is, we take $a=u_{x}$,
$b=u_{y}+u_{x}\,\ln u_{x}$, $p=u_{t}-u\,u_{x}$. Then we consider the linear
combination $\tau=\alpha_{3}-\theta_{1,0}=q^{2}\,(dv-q^{-1}(u\,dt+dx+(q\,w-\ln
u_{x})\,dy)$ and put $q=v_{x}^{-1}$. This yields

$\tau=v_{x}^{-2}\,(dv-u\,v_{x}\,dt-v_{x}\,dx-(w-v_{x}\,\ln u_{x})\,dy).$

The restriction of form $\tau$ on the bundle of sections of the bundle
$\mathbb{R}^{6}\rightarrow\mathbb{R}^{3}$, $(t,x,y,u,v,w)\mapsto(t,x,y)$,
gives the over-determined system of pdes

(4.2) $\left\\{\begin{array}[]{lcl}v_{t}&=&u\,v_{x},\\\ v_{y}&=&w-v_{x}\,\ln
u_{x}.\end{array}\right.$

The integrability condition $(v_{t})_{y}=(v_{y})_{t}$ thereof gives two
equations

(4.3) $u_{tx}=u\,u_{xx}-u_{x}^{2}\,\ln u_{x}-u_{x}\,u_{y}$

and

(4.4) $w_{t}=u\,w_{x}.$

Thus system (4.2) provides a Lax representation of system (4.3), (4.4). We can
find the Lax representation of equation (4.3) by proceeding as follows.
Equation (4.4) is a copy of the first equation in (4.2). We consider $w$ as a
new pseudopotential and include equation (4.4) into the Lax representation for
equation (4.3) by adding the copy $w_{y}=w_{1}-w_{x}\,\ln u_{x}$ of the second
equation from (4.2) with additional function $w_{1}$, and then repeat this
process. In other words, we rename $v=v_{0}$, $w=v_{1}$, then add the sequence
of functions $v_{k}$, $k\geq 2$, and consider the infinite system

(4.5) $\left\\{\begin{array}[]{lcl}v_{k,t}&=&u\,v_{k,x},\\\
v_{k,y}&=&v_{k+1}-v_{k,x}\,\ln u_{x}.\end{array}\qquad k\geq 0.\right.$

The compatibility conditions of this system coincides with equation (4.3).
System (4.5) can be written in the finite form by introducing the function

(4.6)
$r=\mathrm{e}^{-\lambda\,y}\,\sum\limits_{k=0}^{\infty}\lambda^{k}\,v_{k}$

where $\sum\limits_{k=0}^{\infty}\lambda^{k}\,v_{k}$ is a formal series with
respect to a formal parameter $\lambda$. Then we have

(4.7) $\left\\{\begin{array}[]{lcl}r_{t}&=&u\,r_{x},\\\ r_{y}&=&-r_{x}\,\ln
u_{x}.\end{array}\right.$

This system provides a Lax representation for equation (4.3).

Direct computations222We carried out computations of generators of contact
symmetries in the Jets software [2]. give the following statement:

Proposition 1. The contact symmetry algebra of equation (4.3) is generated by
the vector fields

(4.8)
$A_{0}\,\partial_{t}-A_{0}^{\prime}\,y\,\partial_{x}-(A_{0}^{\prime}\,u-A_{0}^{\prime\prime}\,y)\,\partial_{u},\quad
A_{1}\,\partial_{t}-A_{1}^{\prime}\,\partial_{u},\quad
x\,\partial_{x}+y\,\partial_{y}+u\,\partial_{u},\quad\partial_{y},$

where $A_{i}=A_{i}(t)$ are arbitrary smooth functions of $t$. Restricting
these functions to polynomials gives the Lie algebra isomorphic to
$\mathfrak{a}_{2}\ltimes\mathfrak{q}_{1,-1}$. The contact symmetry algebra of
system (4.3), (4.4) is obtained by appending the vector field
$B\,\partial_{v}$ with arbitrary smooth function $B=B(y,v)$ to the vector
fields (4.8).  $\Box$

##  5\. 3D generalized Hunter–Saxton equation

The Lie algebra $\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-1/2}$ admits the
outer derivation $D_{2}(s^{p}\,f(t))=s^{p+1}\,f^{\prime\prime}(t)$, that is,

$D_{2}(s^{p}\otimes V_{k})=\left\\{\begin{array}[]{lcl}k\,(k-1)\,s\otimes
V_{k-2},&\nobreak\leavevmode\hfil&p=0\,\,\,\mathrm{and}\,\,\,k\geq 2,\\\
0,&&p=1\,\,\,\mathrm{or}\,\,\,k\in\\{0,1\\}.\end{array}\right.$

This derivation produces the right extension
$\mathfrak{b}_{1}\ltimes\mathfrak{q}_{1,-1/2}=\langle
Z_{1}\rangle\ltimes(\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-1/2})$ of
$\mathfrak{a}_{0}\ltimes\mathfrak{q}_{1,-1/2}$, where
$[Z_{1},V_{k}]_{-1/2}=D_{2}(V_{k})$ and $[Z_{0},Z_{1}]_{-1/2}=Z_{1}$. Denote
by $\alpha_{1}$ the dual form to $-Z_{1}$, that is, put
$\alpha_{1}(Z_{1})=-1$, $\alpha_{1}(Z_{0})=0$, $\alpha_{1}(V_{k})=0$. Then the
structure equations for $\mathfrak{b}_{1}\ltimes\mathfrak{q}_{1,-1/2}$ acquire
the form

(5.1)
$\left\\{\begin{array}[]{lcl}d\Theta&=&\displaystyle{\Theta_{h_{1}}\wedge\left(\Theta-2\,h_{0}\,\Theta_{h_{0}}\right)+h_{0}\,\alpha_{0}\wedge\Theta_{h_{0}}-h_{0}\,\alpha_{1}\wedge\Theta_{h_{1}h_{1}},}\\\
d\alpha_{0}&=&\displaystyle{0,\phantom{\frac{A}{A}}}\\\
d\alpha_{1}&=&\displaystyle{\alpha_{0}\wedge\alpha_{1}.\phantom{\frac{A}{A}}}\end{array}\right.$

The Lie algebra $\mathfrak{b}_{1}\ltimes\mathfrak{q}_{1,-1/2}$ admits the
sequence of twisted extensions $\mathfrak{b}_{k}\ltimes\mathfrak{q}_{1,-1/2}$,
$k\geq 2$, as it was described when constructing system (3.7). In this section
we need the second extension from this series. Specifically, we have
$[\alpha_{0}\wedge\alpha_{1}]\in
H^{2}_{\alpha_{0}}(\mathfrak{b}_{1}\ltimes\mathfrak{q}_{1,-1/2})$, hence we
append equation

(5.2) $d\alpha_{2}=\alpha_{0}\wedge\alpha_{2}+\alpha_{0}\wedge\alpha_{1},$

to system (5.1) and get the structure equations of
$\mathfrak{b}_{2}\ltimes\mathfrak{q}_{1,-1/2}$. Then
$[\alpha_{1}\wedge\alpha_{2}]\in
H^{2}_{2\alpha_{0}}(\mathfrak{b}_{2}\ltimes\mathfrak{q}_{1,-1/2})$, and the
structure equations of $\mathfrak{b}_{3}\ltimes\mathfrak{q}_{1,-1/2}$ are
obtained by adding equation

(5.3) $d\alpha_{3}=2\,\alpha_{0}\wedge\alpha_{3}+\alpha_{1}\wedge\alpha_{2}.$

to system (5.1), (5.2).

Successively integrating equations (5.1), (5.2), and (5.3) by applying
Frobenius’ theorem, we get

$\qquad\alpha_{0}=\frac{dq}{q},\quad\alpha_{1}=q\,dy,\quad\alpha_{2}=q\,(dw+\ln
q\,\,dy),\quad\alpha_{3}=q^{2}\,(dv-w\,dy),$
$\qquad\theta_{0,0}=a\,dt,\quad\theta_{0,1}=\frac{da}{a}+p_{1}\,dt,\quad\theta_{0,2}=\frac{dp_{1}}{a}+p_{2}\,dt,$
$\qquad\theta_{1,0}=q\,a^{-1}\,(dx+p_{1}\,dt+u\,dt),\quad\theta_{1,1}=q\,a^{-2}\,(du-
p_{1}\,dx+(a\,p_{2}-p_{1}^{2})\,dy+p_{3}\,dt),$

where $q\neq 0$, $a\neq 0$, $t$, $x$, $y$, $u$, $v$, $w$, $p_{1}$, $p_{2}$,
$p_{3}$ are free parameters. By altering notation as $p_{1}=u_{x}$,
$p_{2}=a^{-1}\,(u_{x}^{2}-u_{y})$, and $p_{3}=-u_{t}$ we obtain
$\theta_{1,1}=q\,a^{-2}\,(du-u_{t}\,dt-u_{x}\,dx-u_{y}\,dy)$. Then we consider
the linear combination
$\tau=\alpha_{3}-\theta_{1,0}=q^{2}\,\left(dv-a^{-1}\,q^{-1}\,(u\,dt+dx+(a\,q\,\,w+u_{x})\,dy)\right)$,
put $a=q^{-1}\,v_{x}^{-1}$ and obtain

$\tau=q^{2}\,(dv-u\,v_{x}\,dt-v_{x}\,dx-(w+u_{x}\,v_{x})\,dy).$

Upon restriction to the sections of the bundle
$\mathbb{R}^{6}\rightarrow\mathbb{R}^{3}$, $(t,x,y,u,v,w)\mapsto(t,x,y)$ this
form produces the over-determined system

(5.4) $\left\\{\begin{array}[]{lcl}v_{t}&=&u\,v_{x},\\\
v_{y}&=&w+u_{x}\,v_{x},\end{array}\right.$

which is compatible by virtue of two equations

(5.5) $u_{tx}=u\,u_{xx}-u_{x}^{2}-u_{y}$

and

(5.6) $w_{t}=u\,w_{x}.$

The symmetry reduction of equation (5.5) with respect to $u_{y}=0$ coincides
with the generalized Hunter– Saxton equation [9, 10, 3, 27]

(5.7) $u_{tx}=u\,u_{xx}+\beta\,u_{x}^{2}$

with the special value $\beta=-1$ of parameter $\beta$. Hence (5.5) can be
considered as a three-dimensional generalization of the particular case
$u_{tx}=u\,u_{xx}-u_{x}^{2}$ of equation (5.7).

Likewise to Section 4, we can find the Lax representation of equation (5.5) by
renaming $v=v_{0}$, $w=v_{1}$ and including equation (5.6) in the infinite
system

(5.8) $\left\\{\begin{array}[]{lcl}v_{k,t}&=&u\,v_{k,x},\\\
v_{k,y}&=&v_{k+1}+u_{x}\,v_{k,x},\end{array}\qquad k\geq 0.\right.$

System (5.8) is compatible by virtue of equation (5.5). Series (4.6) allows
one to write (5.8) in the form

(5.9) $\left\\{\begin{array}[]{lcl}r_{t}&=&u\,r_{x},\\\
r_{y}&=&u_{x}\,r_{x}.\end{array}\right.$

We have Proposition 2. The contact symmetry algebra of equation (5.5) admits
generating vector fields

$A_{0}\,\partial_{t}+(A_{0}^{\prime\prime}\,y-A_{0}^{\prime}\,x)\,\partial_{x}-(2\,A_{0}^{\prime}\,u-A_{0}^{\prime\prime}\,x+A_{0}^{\prime\prime\prime}\,y)\,\partial_{u},\quad
A_{1}\,\partial_{x}-A_{1}^{\prime}\,\partial_{u},$
$x\,\partial_{x}+y\,\partial_{y}+u\,\partial_{u},\quad\partial_{y},$

where $A_{i}=A_{i}(t)$ are arbitrary smooth functions of $t$. Restriction of
these functions to $\mathbb{R}[t]$ gives the Lie algebra isomorphic to
$\mathfrak{b}_{2}\ltimes\mathfrak{q}_{1,-1/2}$. The contact symmetry algebra
of system (5.5), (5.6) has additional generating vector field
$B\,\partial_{v}$ with arbitrary smooth function $B=B(y,v)$.  $\Box$

##  6\. Integrable equation associated to
$\mathfrak{a}_{2}\ltimes\mathfrak{q}_{2,-1}$

The structure equations of the Lie algebra
$\mathfrak{a}_{3}\ltimes\mathfrak{q}_{2,-1}$ have the form (4.1), where now
the formal series $\Theta$ is given by formula (3.2) with $N=2$. We take the
same forms $\alpha_{0}$, …, $\alpha_{3}$, and $\theta_{0,0}$ as in Section 4,
and put

$\theta_{1,0}=q\,(db-\ln a\,dy-
p_{1}\,dt),\quad\theta_{2,0}=-q^{2}\,a^{-1}\,(dx+p_{1}\,dy-p_{2}\,dt).$

Then the linear combination

$\theta_{1,0}+\theta_{2,0}+\alpha_{2}=q\,\left(db+dw+\frac{a\,p_{1}+q\,p_{2}}{a}\,dt-\frac{q}{a}\,dx-\frac{a\,(\ln
a-\ln q)+q\,p_{1}}{a}\,dy\right)$

after altering notation $b=u-w$, $q=a\,u_{x}$, $p_{1}=(u_{y}+\ln
u_{x})\,u_{x}^{-1}$, and $p_{2}=-(u_{t}\,u_{x}+u_{y}+\ln u_{x})\,u_{x}^{-2}$
acquires the form $\theta_{1,0}+\theta_{2,0}+\alpha_{2}=q\,(du-u_{t}\,dt-
u_{x}\,dx-u_{y}\,dy)$, while for the form

$\tau=\alpha_{3}+\theta_{2,0}=a^{2}\,u_{x}^{2}\,\left(dv-\frac{1}{a}\,\left(dx+\frac{u_{t}\,u_{x}+u_{y}+\ln
u_{x}}{u_{x}^{2}}\,dt+\frac{a\,u_{x}\,w+u_{y}+\ln
u_{x}}{u_{x}}\,dy\right)\right)$

after renaming $a=v_{x}^{-1}$ we have

$\tau=\frac{u_{x}^{2}}{v_{x}^{2}}\,\left(dv-
v_{x}\,\left(dx+\frac{u_{t}\,u_{x}+u_{y}+\ln
u_{x}}{u_{x}^{2}}\,dt\right)+\frac{u_{x}\,w\,+(u_{y}+\ln
u_{x})\,v_{x}}{u_{x}}\,dy\right).$

Restricting this onto sections of the bundle $(t,x,y,u,v,w)\mapsto(t,x,y)$
produces system

(6.1)
$\left\\{\begin{array}[]{lcl}v_{t}&=&\displaystyle{\frac{u_{t}\,u_{x}+u_{y}+\ln
u_{x}}{u_{x}^{2}}\,v_{x},\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\\\
v_{y}&=&\displaystyle{\frac{u_{y}+\ln
u_{x}}{u_{x}}\,v_{x}+w,\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\end{array}\right.$

which is compatible by virtue of equations

(6.2) $u_{yy}=u_{tx}-\frac{(u_{y}+\ln
u_{x})^{2}+u_{t}\,u_{x}}{u_{x}^{2}}\,u_{xx}+\frac{2\,(u_{y}+\ln
u_{x})-1}{u_{x}}\,u_{xy}$

and

(6.3) $w_{t}=\frac{u_{t}\,u_{x}+u_{y}+\ln u_{x}}{u_{x}^{2}}\,w_{x}.$

The last equation is a copy of the first equation in (6.1). Therefore we can
take $w$ as a new pseudopotential, rename $v=v_{0}$, $w=v_{1}$, add the
infinite sequence of pseudopotentials $v_{j}$, $j\geq 2$, and consider
infinite system

(6.4)
$\left\\{\begin{array}[]{lcl}v_{k,t}&=&\displaystyle{\frac{u_{t}\,u_{x}+u_{y}+\ln
u_{x}}{u_{x}^{2}}\,v_{k,x},\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\\\
v_{k,y}&=&\displaystyle{\frac{u_{y}+\ln
u_{x}}{u_{x}}\,v_{k,x}+v_{k+1},\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\end{array}\right.\qquad
k\geq 0.$

This system is compatible by virtue of equation (6.2) and thus defines a Lax
representation thereof. Introducing series (4.6), we rewrite (6.4) in the form
of another Lax representation

(6.5)
$\left\\{\begin{array}[]{lcl}r_{t}&=&\displaystyle{\frac{u_{t}\,u_{x}+u_{y}+\ln
u_{x}}{u_{x}^{2}}\,r_{x},\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\\\
r_{y}&=&\displaystyle{\frac{u_{y}+\ln
u_{x}}{u_{x}}\,r_{x}\phantom{\frac{\frac{A}{A}}{\frac{A}{A}}}}\end{array}\right.$

for equation (6.2).

Proposition 3. The following vector fields

$A_{0}\,\partial_{t}-\left(A_{0}^{\prime}\,x+\frac{1}{2}\,A_{0}^{\prime\prime}\,y^{2}\right)\,\partial_{x}-A_{0}^{\prime}\,y\,\partial_{u},\quad
A_{1}^{\prime}\,y\,\partial_{x}+A_{1}\,\partial_{u},\quad
A_{2}\,\partial_{x},$
$2\,x\,\partial_{x}+y\,\partial_{y}+(u+y)\,\partial_{u},\quad\partial_{y}.$

with arbitrary smooth functions $A_{i}=A_{i}(t)$ are generators for the
contact symmetry algebra of equation (6.2). Restricting these functions to
polynomials gives the Lie algebra isomorphic to
$\mathfrak{a}_{2}\ltimes\mathfrak{q}_{2,-1}$. The contact symmetry algebra of
system (6.2), (6.3) is obtained by appending the vector field
$B\,\partial_{v}$ with arbitrary smooth function $B=B(y,v)$.  $\Box$

##  7\. The degenerate heavenly equation

In this section we construct a Lie algebra that includes the algebra of
polynomial Hamiltonian vector fields

$\mathfrak{h}=\left\langle
W_{m,n}=n\,t^{m}\,x^{n-1}\,\partial_{t}-m\,t^{m-1}\,x^{n}\,\partial_{x}\,\,|\,\,m,n\in\mathbb{N}_{0},\,\,m^{2}+n^{2}\neq
0\right\rangle$

on $\mathbb{R}^{2}$ as a subalgebra and has a twisted extension that generates
an integrable pde. To find such a Lie algebra we proceed as follows. The Lie
algebra $\mathfrak{h}$ admits the grading $\mathrm{gr}(W_{m,n})=m+n-2$ that
defines the outer derivation $D\colon W_{m,n}\mapsto(m+n-2)\,W_{m,n}$ and the
right extension $\langle Z\rangle\ltimes\mathfrak{h}$ with
$[Z,W_{m,n}]=D(W_{m,n})$. Denote the dual forms to $W_{p,q}$ and $-Z$ as
$\theta_{i,j}$ and $\alpha$, so
$\theta_{i,j}(W_{m,n})=\delta_{i,m}\,\delta_{j,n}$, $\theta_{i,j}(Z)=0$,
$\alpha(W_{m,n})=0$, $\alpha(Z)=-1$. Further, we take $\mathfrak{w}=\langle
V_{k}=\frac{1}{k!}\,y^{k}\,\partial_{y}\,\,|\,\,k\geq 0\rangle$ and consider
the direct sum of Lie algebras

$\widetilde{\mathfrak{h}}=(\langle
Z\rangle\ltimes\mathfrak{h})\oplus\mathfrak{w}.$

Denote by $\omega_{m}$ the dual forms for $V_{k}$, so
$\omega_{m}(V_{k})=\delta_{m,k}$, $\omega_{m}(W_{k,n})=\omega_{m}(Z)=0$, and
$\theta_{m,n}(V_{k})=\alpha(V_{k})=0$. Put

$\Theta=\sum\limits_{m\geq 0,\,n\geq 0,\,m^{2}+n^{2}\neq
0}\,\frac{h_{1}^{m}\,h_{2}^{n}}{m!\,n!}\,\theta_{m,n},\qquad\Omega=\sum\limits_{k\geq
0}\,\frac{h_{3}^{k}}{k!}\,\omega_{k},$

where $h_{i}$ are formal parameters such that $dh_{i}=0$. Then the structure
equations for the Lie algebra $\widetilde{\mathfrak{h}}$ acquire the form

(7.1)
$\left\\{\begin{array}[]{lcl}d\Theta&=&-\Theta_{h_{1}}\wedge\Theta_{h_{2}}-\alpha\wedge\left(h_{1}\,\Theta_{h_{1}}+h_{2}\,\Theta_{h_{2}}-2\,\Theta\right),\\\
d\Omega&=&\Omega_{h_{3}}\wedge\Omega,\\\ d\alpha&=&0.\end{array}\right.$

This system entails $H^{1}(\widetilde{\mathfrak{h}})=\langle\alpha\rangle$ and
$[\theta_{1,0}\wedge\theta_{0,1}]\in
H^{2}_{2\alpha}(\widetilde{\mathfrak{h}})$. Hence the Lie algebra
$\widetilde{\mathfrak{h}}$ admits the twisted extension defined by appending
equation

(7.2) $d\sigma=2\,\alpha\wedge\sigma+\theta_{1,0}\wedge\theta_{0,1}$

to system (7.1).

Applying Frobenius’ theorme and integrating equations (7.1), (7.2) step by
step we obtain

$\theta_{1,0}=a_{11}\,dt+a_{12}\,dx,\quad\theta_{0,1}=a_{21}\,dt+a_{22}\,dx,$
$\alpha=\frac{1}{2}\,\frac{dq}{q},\quad
q=\mathrm{det}\,\left(\begin{array}[]{lcl}a_{11}&\nobreak\leavevmode\hfil&a_{12}\\\
a_{21}&\nobreak\leavevmode\hfil&a_{22}\end{array}\right),$
$\theta_{2,0}=\frac{1}{q}\,(a_{12}\,da_{11}-a_{11}\,da_{12})+b_{1}\,dt+b_{2}\,dx,$
$\theta_{0,2}=\frac{1}{q}\,(a_{22}\,da_{21}-a_{21}\,da_{22})+b_{3}\,dt+b_{4}\,dx,$
$\theta_{1,1}=\frac{1}{2\,q}\,(a_{22}\,da_{11}-a_{11}\,da_{22}+a_{12}\,da_{21}-a_{21}\,da_{12})+b_{5}\,dt+b_{6}\,dx,$
$\omega_{0}=p\,dy,\quad\sigma=q\,(dv-x\,dt),$

where $a_{i,j}$, $b_{1}$, … , $b_{4}$, $p$, $v$ are free parameters such that
$q\neq 0$, $p\neq 0$, and

$b_{5}=\frac{1}{q}\,(-a_{21}\,a_{22}\,b_{1}+a_{21}^{2}\,b_{2}+a_{11}\,a_{12}\,b_{3}-a_{11}^{2}\,b_{4}),$
$b_{6}=\frac{1}{q}\,(-a_{22}^{2}\,b_{1}+a_{21}\,a_{22}\,b_{2}+a_{12}^{2}\,b_{3}-a_{11}\,a_{12}\,b_{4}),$

Put $a_{21}=u\,a_{22}$, $b_{3}=-a_{22}^{2}\,q^{-1}\,u_{t}$,
$b_{4}=-a_{22}^{2}\,q^{-1}\,u_{x}$, $p=-a_{22}^{2}\,q^{-1}\,u_{y}$. This
yields $\theta_{0,2}-\omega_{0}=\frac{a_{22}^{2}}{q}\,(du-u_{t}\,dt-u_{x}\,dx-
u_{y}\,dy)$. Consider the linear combination

$\tau=\sigma+\theta_{1,0}-\textstyle{\frac{1}{2}}\,\omega_{0}$
$\qquad=a_{22}\,(a_{11}-u\,a_{12})\,\left(dv+\frac{dx}{a_{11}-u\,a_{12}}-\frac{(x\,(a_{11}-u\,a_{12})-u)\,dt}{a_{11}-u\,a_{12}}-\frac{u_{y}\,dy}{2\,(a_{11}-u\,a_{12})^{2}}\right).$

By altering notation $a_{11}=u\,a_{12}-v_{x}^{-1}$ we get

$\tau=-\frac{a_{22}}{v_{x}}\,\left(dv-
v_{x}\,dx-(x+u\,v_{x})\,dt-\textstyle{\frac{1}{2}}\,u_{y}\,v_{x}^{2}\,dy\right).$

Then upon restricting form $\tau$ to the sections of the bundle
$\mathbb{R}^{5}\rightarrow\mathbb{R}^{3}$, $(t,x,y,u,v)\mapsto(t,x,y)$, we
obtain the over-determined system

(7.3) $\left\\{\begin{array}[]{lcl}v_{t}&=&x+u\,v_{x},\\\
v_{y}&=&\frac{1}{2}\,u_{y}\,v_{x}^{2}\end{array}\right.$

The compatibility condition of this system $(v_{t})_{y}=(v_{y})_{t}$ holds if
and only

(7.4) $u_{ty}=u\,u_{xy}-2\,u_{x}\,u_{y}.$

Therefore system (7.3) defines a Lax representation for equation (7.4).

We notice that equation (7.4) is invariant with respect to translations along
$x$, while its Lax representation (7.3) does not admit such translations, cf
[21, Th. 4] and [23, Example 7].

Direct computations give

Proposition 4. The contact symmetry algebra of equation (7.4) has generators

$A_{x}\,\partial_{t}-A_{t}\,\partial_{x}+(A_{tt}-2\,u\,A_{tx}+u^{2}\,A_{xx})\,\partial_{u},\quad
B\,\partial_{y},\quad t\,\partial_{t}+x\,\partial_{x},$

where $A=A(t,x)$ and $B=B(y)$ are arbitrary smooth functions of their
arguments. Restriction of these functions to polynomials $A=t^{m}\,x^{n}$,
$B=y^{k}$, $m,n,k\in\mathbb{N}_{0}$, $m^{2}+n^{2}\neq 0$, produces the Lie
algebra isomorphic to $\widetilde{\mathfrak{h}}$.  $\Box$

##  8\. Concluding remarks

In the present paper we have used the method of twisted extensions to derive
new integrable pdes related to some infinite-dimensional Lie algebras. In the
obtained examples as well as in some examples in [22]–[26] the integrable
equations were constructed starting from certain extensions of the Lie
algebras of the Kac–Moody type or the Lie algebra of Hamiltonian vector fields
on $\mathbb{R}^{2}$. In examples of Sections 4, 5, and 6 the symmetry algebras
of the obtained integrable systems turn out to be wider than initial infinite-
dimensional Lie algebras used in the construction, while the specific form of
equations (4.4), (5.6), and (6.3) allowed us to find infinite-component Lax
representations (4.5), (5.8), and (6.4) as well as one-component Lax
representations (4.7), (5.9), and (6.5) for equations (4.3), (5.5), and (6.2),
respectively. The polynomial parts of symmetry algebras of equations (4.3),
(5.5), and (6.2) coincide with the infinite-dimensional Lie algebras whose
twisted extensions were used to construct systems (4.2), (5.4), and (6.1).

We hope further examples will enlighten relations between structure theory of
Lie algebras and integrable pdes. In particular, it is important to address
the following issues in the future research:

  * •

to find other examples of integrable systems related with the Lie algebras of
the Kac–Moody type with small values of $N$,

  * •

to consider extensions of general deformations of the Lie algebras
$\mathbb{R}_{N}[s]\otimes\mathfrak{w}$, cf. [38],

  * •

to construct other integrable systems whose symmetry algebras are extensions
of the Lie algebras of Hamiltonian vector fields,

  * •

to generalize the technique used in the present paper on the Lie–Rinehart
algebras, cf. [28].

## Acknowledgments

I am very grateful to I.S. Krasil′shchik for useful discussions. I thank E.V.
Ferapontov and M.V. Pavlov for important remarks.

## References

  * [1] M.J. Ablowitz, P.A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge, 1991
  * [2] H. Baran, M. Marvan. Jets: A software for differential calculus on jet spaces and diffieties. Available on-line at http://jets.math.slu.cz
  * [3] R. Beals, D.H. Sattinger, J. Szmigielski. Inverse scattering solutions of the Hunter–Saxton equation. Appl. Anal. 78 (2001), 255–269
  * [4] C. P. Boyer, P. Winternitz. Symmetries of self-dual Einstein equations. I. The infinite-dimensional symmetry group and its low-dimensional subgroups. J. Math. Phys. 30 (1989), 1081–1094
  * [5] A. Coley, D. Levi, R. Milson, C. Rogers, P. Winternitz (eds). Bäcklund and Darboux Transformations. The Geometry of Solitons. CRM Proceedings and Lecture Notes, 28, AMS, Providence, 2001
  * [6] D. David, N. Kamran, D. Levi, P. Winternitz. Subalgebras of loop algebras and symmetries of the Kadomtsev–Petviashvili equation. Phys. Rev. Lett. 55 (1985), 2111–2113
  * [7] B. Doubrov, E.V. Ferapontov. On the integrability of symplectic Monge–Ampére equations. J. Geom. Phys. 60 (2010), 1604–1616
  * [8] D.B. Fuks. Cohomology of Infinite-Dimensional Lie algebras. Consultant Bureau: N.Y., 1986
  * [9] J.K. Hunter, R. Saxton. Dynamics of director fields. SIAM J. Appl. Math. 51 (1991), 1498–1521
  * [10] J.K. Hunter, Y.X. Zheng. On a completely integrable nonlinear hyperbolic variational equation. Physica D 79 (1994), 361–386
  * [11] V.G. Kac. Infinite Dimensional Lie Algebras. Cambridge University Press, 1990
  * [12] B.G. Konopelchenko. Nonlinear Integrable Equations. Lecture Notes in Physics, 270, Springer, 1987
  * [13] J. Krasil′shchik, A. Verbovetsky. Geometry of jet spaces and integrable systems// J. Geom. Phys. 61 (2011), 1633–1674
  * [14] J. Krasil′shchik, A. Verbovetsky, R. Vitolo. A unified approach to computation of integrable structures. Acta Appl. Math. 120 (2012), 199–218
  * [15] I.S. Krasil′shchik, A. Verbovetsky, R. Vitolo. The Symmbolic Computation of Integrability Structures for Partial Differential Equations. Springer 2017
  * [16] I.S. Krasil′shchik, A.M. Vinogradov. Nonlocal symmetries and the theory of coverings, Acta Appl. Math. 2 (1984), 79–86
  * [17] I.S. Krasil′shchik, A.M. Vinogradov. Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 15 (1989), 161–209
  * [18] B.S. Kruglikov, O.I. Morozov. SDiff(2) and uniqueness of the Plebański equation. J. Math. Phys. 58:8 (2012), 083506, 11 p.
  * [19] B.S. Kruglikov, O.I. Morozov. Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations. Lett. Math. Phys., 2015, 105, 1703–1723
  * [20] V.B. Matveev, M.A. Salle. Darboux Transformations and Solitons. Springer, 1991
  * [21] O.I. Morozov. Contact integrable extensions of symmetry pseudo-groups and coverings of (2+1) dispersionless integrable equations. J. Geom. Phys. 59 (2009), 1461–1475
  * [22] O.I. Morozov. Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. J. Geom. Phys. 113 (2017), 215–225
  * [23] O.I. Morozov. Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations. J. Geom. Phys. 128 (2018), 20–31
  * [24] O.I. Morozov. Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras. J. Geom. Phys. 143 (2019), 150–163
  * [25] O.I. Morozov. Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein–Weyl structures and the Gibbons–Tsarev equation. Diff. Geom. Appl. 75 (2021), 101740
  * [26] O.I. Morozov. Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation. Diff. Geom. Appl. 76 (2021), 101742
  * [27] O.I. Morozov. Integrability structures of the generalized Hunter–Saxton equation. Analysis and Mathematical Physics (2021) 11:50
  * [28] O.I. Morozov. Integrable partial differential equations and Lie–Rinehart algebras. arXiv:2111.10886
  * [29] S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov. Theory of Solitons. Plenum Press, N.Y., 1984
  * [30] S.P. Novikov. On exotic De-Rham cohomology. Perturbation theory as a spectral sequence. arXiv:math-ph/0201019, 2002
  * [31] S.P. Novikov. On metric-independent exotic homology. Proc. Steklov Inst. Math. 251 (2005), 206–212
  * [32] P.J. Olver. Applications of Lie Groups to Differenial Equations. 2nd Edition, Springer, 1993
  * [33] J. F. Plebański. Some solutions of complex Einstein equations. J. Math. Phys. 16 (1975), 2395–2402
  * [34] C. Rogers, W.F. Shadwick. Bäcklund Transformations and Their Applications. Academic Pres, N.Y., 1982
  * [35] A.M. Vinogradov, I.S. Krasil′shchik (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics [in Russian], Moscow: Factorial, 2005; English transl. prev. ed.: I.S. Krasil′shchik, A.M. Vinogradov (eds.) Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Transl. Math. Monogr., 182, Amer. Math. Soc., Providence, RI, 1999
  * [36] H.D. Wahlquist, F.B. Estabrook. Prolongation structures of nonlinear evolution equations. J. Math. Phys., 16 (1975), 1–7
  * [37] V.E. Zakharov. Integrable systems in multidimensional spaces. Lect. Notes Phys., 153 (1982), 190–216
  * [38] P. Zusmanovich. Deformations of $W_{1}(n)\otimes A$ and modular semisimple Lie algebras with a solvable maximal subalgebra. J. Algebra 268 (2003), 603–635