# Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym
hierarchies and multicomponent Hunter-Saxton equations

Andrey Yu. Konyaev

(November 2021)

###### Abstract

We introduce a natural class of multicomponent local Poisson structures
$\mathcal{P}_{k}+\mathcal{P}_{1}$, where $\mathcal{P}_{1}$ is local Poisson
bracket of order one and $\mathcal{P}_{k}$ is a homogeneous Poisson bracket of
odd order $k$ under assumption that is has Darboux coordinates (Darboux-
Poisson bracket) and non-degenerate. For such brackets we obtain general
formulas in arbitrary coordinates, find normal forms (related to Frobenius
triples) and provide the description of the Casimirs, using purely algebraic
procedure. In two-component case we completely classify such brackets up to
the point transformation. From the description of Casimirs we derive new Harry
Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number
of components. In two component case our HS equation differs from the well-
known HS2 equation.

MSC classes: 37K05, 37K10, 37K25, 53B20, 53B30, 53B50, 35Q53, 35Q35

##  1 Introduction

We start with a motivational example. Recall that Harry Dym (HD) equation is a
non-linear partial differential equation on a function $u(t,x)$, usually
written in compact form

$u_{t}=\Bigg{(}\frac{1}{\sqrt{2u}}\Bigg{)}_{xxx}$ (1)

It was discovered by Kruscal in an unpublished work of Harry Dym (see book
[4]) and rediscovered by Sabatier in his series of papers dedicated to the
classical strings [5]. It plays an important role in study of infinite
dimensional integrable systems and bares a deep connection with KdV, Camassa-
Holm, KP and other integrable hierarchies (see [2], [1] for details and
further references).

Harry Dym equation possesses bihamiltonian structure. Consider the space of
real or complex-valued $C^{\infty}$ functions on $S^{1}$. We denote the
coordinate on the circle as $x\in[0,2\pi]$ and the function itself as $u$,
omitting the dependence in $x$. The $j-$th derivative of $u$ in $x$ is denoted
as $u_{x^{j}}$.

We are interested in the functionals in the form
$\mathcal{H}=\int^{2\pi}_{0}\mathcal{h}\mathrm{d}x$, where density
$\mathcal{h}$ is a finite sum of terms

$\mathcal{h}_{j_{1}\dots j_{s}}u_{x^{j_{1}}}\dots u_{x^{j_{s}}},\quad
j_{1}+\dots+j_{s}=k.$ (2)

Here $\mathcal{h}_{j_{1}\dots j_{s}}$ are smooth (may be complex-valued)
functions of $u$ and, possibly, $x$. The corresponding densities are called
homogeneous differential polynomials of degree $k$.

Consider a pair of Poisson brackets, which act on functionals
$\mathcal{H}_{1}=\int_{0}^{2\pi}\mathcal{h}_{1}\mathrm{d}x$ and
$\mathcal{H}_{2}=\int_{0}^{2\pi}\mathcal{h}_{2}\mathrm{d}x$ as

$\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{1}=\int\limits_{0}^{2\pi}\frac{\delta\mathcal{H}_{1}}{\delta
u}D^{3}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u}\Big{)}\mathrm{d}x,\quad\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{2}=\int\limits_{0}^{2\pi}\Big{(}2u\frac{\delta\mathcal{H}_{1}}{\delta
u}D\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u}\Big{)}+u_{x}\frac{\delta\mathcal{H}_{1}}{\delta
u}\frac{\delta\mathcal{H}_{2}}{\delta u}\Big{)}\mathrm{d}x.$ (3)

Here $D$ is total derivative in variable $x$ and $\frac{\delta}{\delta u}$ is
variational derivative in field variable $u$. Brackets (3) are compatible in a
usual sense: their linear combination with arbitrary constant coefficients is
again a Poisson bracket.

The equation (1) is Hamiltonian with respect to both brackets. Moreover, there
exists a sequence of functionals
$\mathcal{H}_{i}=\int_{0}^{2\pi}\mathcal{h}_{i}\mathrm{d}x,i\geq 1$, such that
the corresponding flows have the form

$\displaystyle u_{t_{0}}$
$\displaystyle=2uD\Big{(}\frac{\delta\mathcal{H}_{1}}{\delta
u}\Big{)}+u_{x}\frac{\delta\mathcal{H}_{1}}{\delta u}=0,$ (4) $\displaystyle
u_{t_{i}}$ $\displaystyle=2uD\Big{(}\frac{\delta\mathcal{H}_{i+1}}{\delta
u}\Big{)}+u_{x}\frac{\delta\mathcal{H}_{i+1}}{\delta
u}=D^{3}\Big{(}\frac{\delta\mathcal{H}_{i}}{\delta u}\Big{)},\quad i\geq 1.$

The flows (4) pairwise commute and Harry Dym equation (1) is $u_{t_{1}}$ for
the functional $\mathcal{H}_{1}=\int^{2\pi}_{0}\sqrt{2u}\mathrm{d}x$.
Functionals $\mathcal{H}_{i},i\geq 1$ are common conservation laws for all the
equations (4). Usually the term hierarchy refers to the collection of
$u_{t_{i}}$, but we use it for the collection of $\mathcal{H}_{i}$ as well.

The densities $\mathcal{h}_{i}$ for functionals $\mathcal{H}_{i}$ can be found
by purely algebraic means, without solving any differential equations.
Consider the identity

$u=\frac{1}{2}\mathrm{v}^{2}+\mathrm{v}_{x}.$ (5)

We search the solution in a form of a series

$\mathrm{v}=\mathcal{v}_{1}+\mathcal{v}_{2}+\mathcal{v}_{3}+\dots,$ (6)

where $\mathrm{v}_{i}$ is either zero or a homogeneous differential polynomial
of degree $i-1$. Substituting $\mathrm{v}$ into (5) yields the recursion
formulas for the components of the series

$\displaystyle\mathcal{v}^{2}_{1}$ $\displaystyle=2u,$
$\displaystyle\mathcal{v}_{i+1}$
$\displaystyle=-\frac{1}{\mathcal{v}_{1}}\Bigg{(}\,\frac{1}{2}\sum\limits_{j=2}^{i}\mathcal{v}_{j}\mathcal{v}_{i+2-j}+(\mathcal{v}_{i})_{x}\Bigg{)},\quad
i\geq 1.$

The series is uniquely defined by the solution of the first equation, that is
the choice of either $\mathcal{v}_{1}=\sqrt{2u}$ or
$\mathcal{v}_{1}=-\sqrt{2u}$. Taking $\mathcal{h}_{i}=\mathcal{v}_{2i-1}$ we
get a collection of functionals $\mathcal{H}_{i}$. This is exactly the Harry
Dym hierarchy.

We can try and continue hierarchy for negative $i$. For $i=-1$ and $i=-2$ we
get

$\displaystyle u_{t_{0}}$
$\displaystyle=D^{3}\Big{(}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
u}\Big{)}=0,$ (7) $\displaystyle u_{t_{-1}}$
$\displaystyle=D^{3}\Big{(}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta
u}\Big{)}=2uD\Big{(}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
u}\Big{)}+u_{x}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta u},$ $\displaystyle
u_{t_{-2}}$
$\displaystyle=2uD\Big{(}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta
u}\Big{)}+u_{x}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u}$

We take $\bar{\mathcal{H}}_{0}=\int^{2\pi}_{0}u\mathrm{d}x$. Introducing new
variable $u=v_{xx}$ we get $\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u}=v$.
After renaming $t_{-2}$ as $t$ the third equation in (7) takes form
$v_{xxt}=2v_{xx}v_{x}+v_{xxx}v$. Integrating it by $x$ and taking the
integration constant to be zero, we arrive to the Hunter-Saxton equation

$v_{xt}=v_{xx}v+\frac{1}{2}v_{x}^{2}.$ (8)

This equation originally appeared in the study of nematic liquid crystals.

All the mentioned well-known results admit a natural interpretation in terms
of inhomogeneous Poisson brackets. Consider a sum of the brackets (3), which
is again a Poisson bracket (actually, it is one of KdV brackets)

$\displaystyle\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}$
$\displaystyle=\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{1}+\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{2}=$
(9)
$\displaystyle=\int\limits_{0}^{2\pi}\Bigg{(}\frac{\delta\mathcal{H}_{1}}{\delta
u}D^{3}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u}\Big{)}+2u\frac{\delta\mathcal{H}_{1}}{\delta
u}D\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u}\Big{)}+u_{x}\frac{\delta\mathcal{H}_{1}}{\delta
u}\frac{\delta\mathcal{H}_{2}}{\delta u}\Bigg{)}\mathrm{d}x.$

Using Lemma 6.4 from section 6 one can show that that Poisson bracket (9) has
no Casimirs in the category of differential polynomials, except constants.

At the same time one can extend the Poisson bracket (9) onto the space of the
series of differential polynomials in the form (6). We treat such series as
density, which depends on all derivatives $u_{x^{j}}$. The corresponding
functional is

$\mathrm{W}=\int\limits_{0}^{2\pi}\mathrm{w}\mathrm{d}x=\sum\limits_{i=1}^{\infty}\int\limits_{0}^{2\pi}\mathcal{w}_{i}\mathrm{d}x=\sum\limits_{i=1}^{\infty}\mathcal{W}_{i}.$

For Harry Dym hierarchy
$\mathcal{H}_{i}=\int^{2\pi}_{0}\mathcal{h}_{i}\mathrm{d}x$ consider the
functional $\mathrm{H}=\int_{0}^{2\pi}\mathrm{h}\mathrm{d}x$ for density
$\mathrm{h}=\sum_{i=1}(-1)^{i+1}\mathcal{h}_{i}$. Then for arbitrary
functional $\mathrm{W}=\int_{0}^{2\pi}\mathrm{w}\mathrm{d}x$ the conditions
(4) yield

$\\{\mathrm{W},\mathrm{H}\\}=0.$

In other words, the Harry Dym hierarchy defines the Casimirs of the
inhomogeneous Poisson bracket. This implies, that the geometry of such bracket
(the description of bracket and Casimirs) contains all the necessary
information, that allows one to write the equations in any given coordinates.

We apply this idea in case of arbitrary $n$. To do that we introduce a natural
class of inhomogeneous Poisson brackets, for which the geometry can be
completely described. This, in turn, leads to the natural (and new!)
generalizations of Harry Dym hierarchies and Hunter-Saxton equations. The
brackets itself are described by a purely geometric object: a pair of flat
metrics with certain compatibility conditions. The Casimirs and corresponding
hierarchies are found by purely algebraic means. In our work we use the
language of Dubrovin-Novikov brackets as the most general one. The further
results, associated with compatible inhomogeneous brackets of the given type
are given in [3].

First, we recall some basic definitions and results of the theory of Dubrovin-
Novikov brackets (for more detailed exposure on the subject we refer the
reader to [6], [14], [15], [16]).

Let $\mathrm{M}^{n}$ be the $n-$dimensional ball. Denote
$\Omega\mathrm{M}^{n}$ to be a space of maps from $S^{1}\to\mathrm{M}^{n}$.
The functionals $\mathcal{H}:\Omega\mathrm{M}^{n}\to\mathbb{K}$ (for
$\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$), we are interested in, are written
(same as for $n=1$) in the form
$\mathcal{H}=\int^{2\pi}_{0}\mathcal{h}\mathrm{d}x$. Here $\mathcal{h}$ is a
differential polynomial in $n$ field variables.

The local homogeneous Poisson bracket of order $k$ is a bilinear skew-
symmetric operation, which acts on functional
$\mathcal{H}_{1}=\int_{0}^{2\pi}\mathcal{h}_{1}\mathrm{d}x$ and
$\mathcal{H}_{2}=\int_{0}^{2\pi}\mathcal{h}_{2}\mathrm{d}x$ as

$\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}=\int\limits^{2\pi}_{0}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}h^{\alpha\beta}D^{k}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Big{)}\mathrm{d}x+\int\limits^{\infty}_{0}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}\sum\limits_{j=1}^{k}\mathcal{P}^{\alpha\beta}_{\mathrm{j}}D^{k-j}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Big{)}\mathrm{d}x$ (10)

and satisfies the Jacobi identity. The entries of matrix $h^{\alpha\beta}$ are
the functions of field variables $u^{\alpha}$ only. The entries of matrices
$\mathcal{P}^{\alpha\beta}_{\mathrm{j}}$ are homogeneous differential
polynomials of degree $j$. The skew-symmetry condition and Jacobi identity
provide differential conditions on components of
$\mathcal{P}^{\alpha\beta}_{\mathrm{j}}$.

The symbolic expression

$\mathcal{P}^{\alpha\beta}=h^{\alpha\beta}D^{k}+\sum\limits_{j=1}^{k}\mathcal{P}^{\alpha\beta}_{\mathrm{j}}D^{k-j}$
(11)

is called a Hamiltonian operator. For Poisson bracket (10) it plays the same
role as Poisson tensor plays for finite-dimensional Poisson bracket.

For a given functional $\mathcal{H}=\int^{2\pi}_{0}\mathcal{h}\mathrm{d}x$ the
corresponding Hamiltonian flow is defined as

$u^{\alpha}_{t}=\mathcal{P}^{\alpha q}\frac{\delta\mathcal{H}}{\delta
u^{q}}=h^{\alpha q}D^{k}\Big{(}\frac{\delta\mathcal{H}}{\delta
u^{q}}\Big{)}+\sum\limits_{j=1}^{k}\mathcal{P}^{\alpha
q}_{\mathrm{j}}D^{k-j}\Big{(}\frac{\delta\mathcal{H}}{\delta u^{q}}\Big{)}.$

For arbitrary test functional
$\mathcal{H}_{1}=\int^{2\pi}_{0}\mathcal{h}_{1}\mathrm{d}x$ it satisfies the
obvious relation
$\partial_{t}\mathcal{H}_{1}=\\{\mathcal{H}_{1},\mathcal{H}\\}$.

Under the coordinate change
$\bar{u}(u)=(\bar{u}^{1}(u^{1},\dots,u^{n}),\dots,\bar{u}^{n}(u^{1},\dots,u^{n}))$
the components of the Hamiltonian operator are transformed as (formula 3.2
[6]):

$\bar{h}^{\alpha\beta}=h^{pq}\frac{\partial\bar{u}^{\alpha}}{\partial
u^{p}}\frac{\partial\bar{u}^{\beta}}{\partial
u^{q}},\quad\bar{\mathcal{P}}_{\mathrm{j}}^{\alpha\beta}=\sum\limits_{i=j}^{k}\binom{j}{i}\mathcal{P}_{\mathrm{i}}^{pq}\frac{\partial\bar{u}^{\alpha}}{\partial
u^{p}}D^{j-i}\Big{[}\frac{\partial\bar{u}^{\beta}}{\partial u^{q}}\Big{]}.$

Here $\binom{j}{i}$ are binomial coefficients, $\bar{h}^{\alpha\beta}$ and
$\bar{\mathcal{P}}_{\mathrm{j}}^{\alpha\beta}$ stand for the coefficients of
Hamiltonian operator in coordinates $\bar{u}^{1},\dots,\bar{u}^{n}$.

As it was mentioned earlier, the term $h^{\alpha\beta}$ does not depend on
differential coordinates and under the coordinate transformation behaves like
a contrvariant tensor of type $(2,0)$. Due to skew-symmetry condition of the
Poisson bracket (formula 3.3, [6]) this tensor is symmetric for odd $k$ and
skew-symmetric for even $k$. The Hamiltonian operator is called non-degenerate
if $\operatorname{det}h\neq 0$.

The analog of (9) is an inhomogeneous Poisson bracket, for which its
Hamiltonian operator is a sum of two homogeneous parts $\mathcal{A}$ and
$\mathcal{B}$ of orders $1$ and $3$ respectively. Due to Theorem 7.1 in [6]
both $\mathcal{A}$ and $\mathcal{B}$ are Hamiltonian operators on their own.
Thus, by definition, we have a pair of compatible Poisson brackets.

The local homogeneous Poisson bracket of order $k$ is Darboux-Poisson if there
exist coordinates $u^{1},\dots,u^{n}$, such that the Poisson bracket takes
form

$\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}=\int_{0}^{\infty}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}h^{\alpha\beta}D^{k}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Big{)}\mathrm{d}x.$

Following [6] we call such coordinates Darboux coordinates. The geometry of
homogeneous Poisson brackets of order $k=3$ was investigated by a number of
authors ([6], [11], [12], [13]). In our work we assume that the bracket of
order $k>1$ is Darboux-Poisson.

The non-degenerate homogeneous Poisson brackets of order one are called
Poisson brackets of hydrodynamic type. The physicists used this type of
brackets for a long time: Landau, for example, used them without any specific
name in formulation of his theory of the superfluidity for hellium (formula
1.4 in [10]).

The geometric theory of these objects was developed by Dubrovin and Novikov in
the 1980s ([7], [8], see also [9]). They showed, in particular, that these
Poisson brackets always have Darboux coordinates and, thus, are in one-to-one
correspondence with flat metrics. We will denote the Hamiltonian operator of
non-degenerate homogeneous Poisson bracket of order one as $\mathcal{A}_{g}$,
where $g$ is the corresponding flat contrvariant metric.

The paper is organized as follows: in the next section 2 we formulate the main
results and provide important examples. The Theorem 2.1 provides the formula
for Darboux-Poisson brackets in local coordinates for arbitrary $k$. It
generalizes the famous formula by Dobrovin and Novikov. In particular, it
implies that for odd $k$ such operator is uniquely defined by a flat metric.

Theorem 2.2 yields normal form for inhomogeneous Poisson brackets with the odd
term of order $k\geq 3$ being homogeneous Darboux-Poisson bracket. For $k=3$
the statement of the Theorem 2.2 was independently obtained by P. Lorenzoni
and R. Vitolo in their unpublished paper.

For $k=3$ Theorem 2.2 establishes the relation between general geometric
approach and well-known algebraic approach, which appeared in original works
by Balinskii and Novikov [21]. It implies that any such bracket corresponds to
a Frobenius triple (see the exact definitions below) — that is a commutative
associative algebra with a pair of non-degenerate invariant forms. This
implies, that in general the description of such brackets is a lost cause.
Yet, in small dimension it is possible. Theorem 2.3 provides the
classification of Darboux-Poisson brackets in dimension two. This
classification contains the inhomogeneous brackets of many famous integrable
systems.

The last Theorem 2.4 provides an algebraic procedure to construct the
conservation laws of such system in an implicit form. Similar to the case of
dimension one we get an equation, bares similarities to both Balinskii-Novikov
formula and Miura formula. This identity provides the recursion relation,
which can be solved.

In the end of the section we provide the construction of Hunter-Saxton
equation in case of arbitrary inhomogeneous Darboux-Poisson bracket with minor
non-degeneracy conditions (formula (31)). We also describe two important
examples, for which we write all the formulas explicitly. We compare them with
the cases, known in the literature, and show, that they are, indeed, new.

All the proofs are provided in sections 3, 4, 5, 6. The Appendix contains some
important algebraic observations, which (to keep work self-sufficient) are
provided with proofs.

##  2 Main results of the paper

We start with general result on Darboux-Poisson brackets.

######  Theorem 2.1.

Consider homogeneous a non-degenerate Darboux-Poisson bracket of order $k$ in
arbitrary coordinates $u^{1},\dots,u^{n}$ with Hamiltonian operator

$\mathcal{P}^{\alpha\beta}=h^{\alpha\beta}D^{k}+\sum\limits_{j=1}^{k}\mathcal{P}^{\alpha\beta}_{\mathrm{j}}D^{k-j}.$
(12)

Recall, that $h^{\alpha\beta}$ is non-degenerate contrvariant tensor of rank
two. We have

  1. 1.

The term $P_{1}^{q\beta}$ has the form

$\mathcal{P}_{1}^{\alpha\beta}=-kh^{\alpha q}\Gamma^{\beta}_{qs}u^{s}_{x},$
(13)

where $\Gamma^{\beta}_{qs}$ is flat symmetric connection.

  2. 2.

The contrvariant tensor $h^{\alpha\beta}$ is parallel along
$\Gamma^{\beta}_{qs}$ and the flat coordinates of the connection are exactly
the Darboux coordinates of the Darboux-Poisson bracket (12).

  3. 3.

For $j\geq 1$ define matrices $\tau(j)^{\beta}_{q}$ using recursion formula

$\tau(1)^{\beta}_{q}=-\Gamma^{\beta}_{qs}u^{s}_{x},\quad\tau(j+1)^{\beta}_{q}=\tau(1)^{m}_{q}\tau(j)^{\beta}_{m}+D\big{(}\tau(j)^{\beta}_{q}\big{)}.$
(14)

Then the rest of the coefficients of $\mathcal{P}$ has the form

$\mathcal{P}^{\alpha\beta}_{\mathrm{j}}=\binom{j}{k}h^{\alpha
q}\tau(j)^{\beta}_{q},\quad j\geq 2.$ (15)

The inverse is also true: for a given collection of data

  1. 1.

non-degenerate symmetric (skew-symmetric) contrvariant tensor
$h^{\alpha\beta}$ of type $(2,0)$;

  2. 2.

flat symmetric connection $\Gamma^{\beta}_{qs}$ with condition $\nabla
h^{\alpha\beta}=0$;

  3. 3.

odd (even) natural number $k$;

the formulas (13), (15), (14) and (12) define homogeneous non-degenerate
Hamiltonian operator of Darboux-Poisson bracket of order $k$ in given
coordinates.

######  Remark 2.1.

For even $k$ non-degenerate skew-symmetric tensor $h^{\alpha\beta}$ is
actually the inverse of symplectic structure and $\Gamma^{\beta}_{rs}$ is flat
symplectic connection. The corresponding pair defines the so called structure
of flat Fedosov manifold [17], [18], which is the main ingredient of Fedosov
quantization procedure. To study the analogs of Harry Dym equations in this
case is an interesting problem, but it is out of the scope of this paper.
$\blacksquare$

In case of odd $k$ the Christoffel symbols $\Gamma^{\beta}_{rs}$ define Levi-
Civita connection of corresponding flat metric $h$. Thus, all the ingredients
of the Darboux-Poisson bracket are defined by single flat metric. So for
Poisson bracket of odd order $k>1$ we adopt the notion $\mathcal{B}_{h}$,
where $h$ is corresponding flat metric.

Note, that for $k=1$, Theorem 2.1 yields the classical formulas from [7] by
Dubrovin and Novikov

$\mathcal{A}^{\alpha\beta}_{g}=g^{\alpha\beta}D-g^{\alpha
q}\Gamma^{\beta}_{qs}u^{s}_{x},$ (16)

with only change $h^{\alpha\beta}\to g^{\alpha\beta}$. For $k=3$ the formula
for components of the Hamiltonian operator of Darboux-Poisson bracket
$\mathcal{B}_{h}$ is

$\displaystyle\mathcal{B}^{\alpha\beta}_{h}$
$\displaystyle=h^{\alpha\beta}D^{3}-3h^{\alpha
q}\bar{\Gamma}^{\beta}_{qs}u^{s}_{x}D^{2}+$ (17)
$\displaystyle+3\Bigg{(}h^{\alpha
q}\Big{(}\bar{\Gamma}^{p}_{qs}\bar{\Gamma}^{\beta}_{pr}-\frac{\partial\bar{\Gamma}^{\beta}_{qs}}{\partial
u^{r}}\Big{)}u^{s}_{x}u^{r}_{x}-h^{\alpha
q}\bar{\Gamma}^{\beta}_{qs}u^{s}_{x^{2}}\Bigg{)}D+$
$\displaystyle+\Bigg{(}h^{\alpha
q}\Big{(}2\bar{\Gamma}^{a}_{qs}\frac{\partial\bar{\Gamma}^{\beta}_{ar}}{\partial
u^{p}}+\frac{\partial\bar{\Gamma}^{a}_{qs}}{\partial
u^{r}}\bar{\Gamma}^{\beta}_{ap}-\bar{\Gamma}^{a}_{qs}\bar{\Gamma}^{b}_{ar}\bar{\Gamma}^{\beta}_{bp}-\frac{\partial^{2}\bar{\Gamma}^{\beta}_{qs}}{\partial
u^{r}\partial u^{p}}\Big{)}u^{s}_{x}u^{r}_{x}u^{p}_{x}+$
$\displaystyle+h^{\alpha
q}\Big{(}2\bar{\Gamma}^{a}_{qs}\bar{\Gamma}_{ar}^{\beta}+\bar{\Gamma}^{a}_{qr}\bar{\Gamma}^{\beta}_{as}-2\frac{\partial\bar{\Gamma}^{\beta}_{qr}}{\partial
u^{s}}-\frac{\partial\bar{\Gamma}^{\beta}_{qs}}{\partial
u^{r}}\Big{)}u^{s}_{x}u^{r}_{x^{2}}-h^{\alpha
q}\bar{\Gamma}^{\beta}_{qs}u^{s}_{x^{3}}\Bigg{)}.$

Here $\bar{\Gamma}^{\beta}_{qs}$ are Christoffel symbols of Levi-Civita
connection of $h$ (we use notation with bar to distinguish formulas (16) and
(17)). The inhomogeneous Poisson brackets $\mathcal{P}$

$\mathcal{P}=\mathcal{A}_{g}+\mathcal{B}_{h}$

is, thus, determined by a pair of flat metrics $g$ and $h$. Of course, these
metrics are not arbitrary: the compatibility of $\mathcal{A}_{g}$ and
$\mathcal{B}_{g}$ puts strong conditions on them.

######  Theorem 2.2.

Let $\mathcal{A}_{g}$ be the Hamiltonian operator of homogeneous non-
degenerate Poisson bracket of order one and $\mathcal{B}_{h}$ be the
Hamiltonian operator of homogeneous non-degenerate Darboux-Poisson bracket of
odd order $k$. The formula

$\mathcal{P}^{\alpha\beta}=\mathcal{A}^{\alpha\beta}_{g}+\mathcal{B}^{\alpha\beta}_{h}$
(18)

defines the Hamiltonian operator of inhomogeneous Poisson bracket of type
$1+k$ (equivalently, $\mathcal{A}_{g}$ and $\mathcal{B}_{h}$ are compatible),
if and only if

  * $k\geq 5$:

There exist coordinates $u^{1},\dots,u^{n}$, in which both metrics $h$ and $g$
are in constant form. The Poisson bracket in these coordinates takes form

$\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}=\int\limits_{0}^{2\pi}\Bigg{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}h^{\alpha\beta}D^{k}\Big{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\beta}}\Big{)}+\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}g^{\alpha\beta}D\Big{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\beta}}\Big{)}\Bigg{)}\mathrm{d}x.$ (19)

  * $k=3$:

In coordinates $u^{1},\dots,u^{n}$, where $h$ has constant entries
$h^{\alpha\beta}$, the metric $g$ has the form

$g^{\alpha\beta}=2(b^{\alpha\beta}+a^{\alpha\beta}_{s}u^{s}).$

The constants $h^{\alpha\beta},b^{\alpha\beta},a^{\alpha\beta}_{s}$ are
symmetric in upper indices and satisfy the conditions

$\displaystyle
a^{\alpha\beta}_{q}b^{q\gamma}=a^{\gamma\beta}_{q}b^{q\alpha},\quad
a^{\alpha\beta}_{q}a^{q\gamma}_{s}=a^{\gamma\beta}_{q}a^{q\alpha}_{s},\quad
a^{\alpha\beta}_{q}h^{q\gamma}=a^{\gamma\beta}_{q}h^{q\alpha}.$ (20)

The Poisson bracket in these coordinates takes form

$\displaystyle\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}=\int\limits_{0}^{2\pi}\Bigg{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}h^{\alpha\beta}D^{3}\Big{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\beta}}\Big{)}+2\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}(b^{\alpha\beta}+a^{\alpha\beta}_{s}u^{s})D\Big{(}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\beta}}\Big{)}+\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}a^{\alpha\beta}_{s}u^{s}_{x}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Bigg{)}\mathrm{d}x.$ (21)

######  Corollary 2.1.

Consider a pair of flat metrics $g^{\alpha\beta}$ and $h^{\alpha\beta}$ with
Christoffel symbols of corresponding metrics being $\Gamma^{\alpha\beta}_{s}$
and $\bar{\Gamma}^{\alpha\beta}_{s}$ respectively and let
$S^{\alpha\beta}_{s}=g^{\alpha
q}\big{(}\Gamma^{\beta}_{qs}-\bar{\Gamma}^{\beta}_{qs}\big{)}$. Denote
$\mathcal{A}_{g}$ and $\mathcal{B}_{h}$ to be the same Hamiltonian operators
as in the statement of Theorem 2.2. The formula (18) defines the Hamiltonian
operator of inhomogeneous Poisson bracket of type $1+k$ if and only if

  * $k\geq 5$:

Tensor $S^{\alpha\beta}_{s}$ vanishes

  * $k=3$:

The tensor $S^{\alpha\beta}_{s}$ is symmetric in upper indices and satisfies
conditions

$\displaystyle\bar{\nabla}_{q}S^{\alpha\beta}_{s}=0,\quad
S^{\alpha\beta}_{q}S^{q\gamma}_{s}=S^{\gamma\beta}_{q}S^{q\alpha}_{s},\quad
S^{\alpha\beta}_{q}h^{q\gamma}_{s}=S^{\gamma\beta}_{q}h^{q\alpha}_{s},\quad
S^{\alpha\beta}_{q}b^{q\gamma}_{s}=S^{\gamma\beta}_{q}b^{q\alpha}_{s}.$ (22)

In flat coordinates of $h^{\alpha\beta}$, the constants $a^{\alpha\beta}_{s}$
from formula (21) are related to tensor $S^{\alpha\beta}_{s}$ as
$a^{\alpha\beta}_{s}=-S^{\alpha\beta}_{s}$.

The first statement of Theorem 2.2 shows, that in our line of investigation
there are no non-trivial higher order analogs (for example, with fifth order
derivative in $x$) of Harry Dym equation. The Hamiltonian operator for $k=3$
in given dimension is defined by a finite collection of constants
$a^{\alpha\beta}_{s},b^{\alpha\beta},h^{\alpha\beta}$. Let us explore the
algebraic properties of these constants.

First, recall that flat coordinate system for $h$ is defined up to an affine
transformation. In case of $k=3$ the translation by vector
$c=(c^{1},\dots,c^{n})$ changes $b^{\alpha\beta}$ into
$\bar{b}^{\alpha\beta}=b^{\alpha\beta}-a^{\alpha\beta}_{s}c^{s}$.

Let $\mathrm{p}$ be a coordinate origin of the coordinate system
$u^{1},\dots,u^{n}$, in which the inhomogeneou Poisson bracket is in the form
(21). Restricting ourselves to the coordinate changes that preserve point
$\mathrm{p}$, we observe that the constants $a^{\alpha\beta}_{s}$ behave as
tensor of type $(2,1)$, while $h^{\alpha\beta},b^{\alpha\beta}$ behave as
tensors of type $(2,0)$.

Consider a commutative associative algebra $\mathfrak{a},\star$ over
$\mathbb{R}$ or $\mathbb{C}$. The symmetric bilinear form $h$ is invariant
form if

$h(\xi\star\eta,\zeta)=h(\xi,\eta\star\zeta),\quad\forall\xi,\eta,\zeta\in\mathfrak{a}.$
(23)

If an invariant form is non-degenerate, then it is called Frobenius form. A
Frobenius triple $(\mathfrak{a},b,h)$ is a commutative associative algebra
$\mathfrak{a}$, equipped with two Frobenius forms $b$ and $h$.

The formulas (20) and (23) imply that constants
$a^{\alpha\beta}_{s},b^{\alpha\beta},h^{\alpha\beta}$ are just components of
some Frobenius triple, written in a given basis of $T^{*}\mathrm{M}$.
Moreover, the coordinate change that preserves coordinate origin $\mathrm{p}$
does not change the triple. So, in algebraic terms, the Theorem 2.2
establishes one-to-one correspondence between inhomogeneous Hamiltonian
operators of the form (18) around given point $\mathrm{p}\in\mathrm{M}$ and
Frobenius triples on $T^{*}_{\mathrm{p}}\mathrm{M}^{n}$.

######  Remark 2.2.

In case of degenerate $h^{\alpha\beta}$ even if the Poisson bracket is in the
form (21) the constants $a^{\alpha\beta}_{s}$ do not necessarily define
commutative associative algebra. The following is a Hamiltonian operator of
inhomogeneous Poisson bracket ([22], the case of algebra N4)

$\mathcal{P}^{\alpha\beta}=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)D^{3}+\left(\begin{array}[]{cc}2u^{1}&u^{2}\\\
u^{2}&0\end{array}\right)D+\left(\begin{array}[]{cc}u^{1}_{x}&0\\\
u^{2}_{x}&0\end{array}\right).$

The corresponding algebra is non-commutative, but still associative (it is, in
fact, an associative Novikov algebra). At the same time coordinate change
$\bar{u}^{2}=u^{1},\bar{u}^{1}=(u^{2})^{2}$ transforms the Hamiltonian
operator of the Poisson bracket into the form

$\bar{\mathcal{P}}^{\alpha\beta}=\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right)D^{3}+\left(\begin{array}[]{cc}0&2\bar{u}^{1}\\\
2\bar{u}^{1}&2\bar{u}^{2}\end{array}\right)D+\left(\begin{array}[]{cc}0&\bar{u}^{1}_{x}\\\
\bar{u}^{1}_{x}&\bar{u}^{2}_{x}\end{array}\right).$

Now the algebra, corresponding to $g^{\alpha\beta}$ is commutative and
associative. This Hamiltonian operator can be obtained as a limit $h_{1}\to 0$
in family $(31)$ (see Theorem 2.3). $\blacksquare$

Theorem 2.2 implies, that for odd $k>3$ the inhomogeneous Poisson bracket in
the form (18) are in one-to-one correspondence with a pair of symmetric forms
$g,h$. The explicit description all such pairs for given dimension $n$ is
cumbersome, but possible (see, for example, [19]).

In case of $k=3$ the problem of classification of inhomogeneous Hamiltonian
operators in the form (18) contains the problem of classification commutative
associative algebras with Frobenius forms. For arbitrary signature of the
forms this problem seems to have no solution in general.

Yet for any given dimension the classification is possible. In dimension two
up to isomorphism there are five non-trivial associative commutative algebras,
which we denote $\mathfrak{a}_{1},\dots,\mathfrak{a}_{5}$. The trivial algebra
we denote as $\mathfrak{a}_{0}$. In basis $\eta^{1},\eta^{2}$ (for the
consistency of the notation we assume that the coordinates on algebra have
lower indices) the non-zero structure relations for each of them are

$\displaystyle\mathfrak{a}_{1}:$
$\displaystyle\quad\quad\eta^{2}\star\eta^{2}=\eta^{2},$ (24)
$\displaystyle\mathfrak{a}_{2}:$
$\displaystyle\quad\quad\eta^{2}\star\eta^{2}=\eta^{1},$
$\displaystyle\mathfrak{a}_{3}:$
$\displaystyle\quad\quad\eta^{2}\star\eta^{2}=\eta^{2},\quad\eta^{1}\star\eta^{2}=\eta^{1},$
$\displaystyle\mathfrak{a}_{4}:$
$\displaystyle\quad\quad\eta^{1}\star\eta^{1}=\eta^{1},\quad\eta^{2}\star\eta^{2}=\eta^{2},$
$\displaystyle\mathfrak{a}_{5}:$
$\displaystyle\quad\quad\eta^{2}\star\eta^{2}=\eta^{2},\quad\eta^{1}\star\eta^{1}=-\eta^{2},\quad\eta^{1}\star\eta^{2}=\eta^{1}.$

The following Theorem describes the inhomogeneous Hamiltonian operators in
dimension two.

######  Theorem 2.3.

Consider point $\mathrm{p}\in\mathrm{M}^{2}$. Every inhomogeneous Hamiltonian
operator $\mathcal{P}$ of the form (18) for odd $k$ can be brought around
point $\mathrm{p}$ to exactly one normal form from the list below. The
different values of parameters $h_{1},h_{2},b_{1},b_{2}$ yield non-equivalent
normal forms, the canonical coordinates $u^{1},u^{2}$ are centered at
$\mathrm{p}$:

  * $(01)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)D$

  * $(02)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+\left(\begin{array}[]{cc}-1&0\\\ 0&1\\\
\end{array}\right)D$

  * $(03)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+\left(\begin{array}[]{cc}-1&0\\\ 0&-1\\\
\end{array}\right)D$

  * $(04)$:

$\left(\begin{array}[]{cc}0&h_{1}\\\ h_{1}&1\\\
\end{array}\right)D^{3}+\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)D$

  * $(05)$:

$\left(\begin{array}[]{cc}-h_{2}&h_{1}\\\ h_{1}&h_{2}\\\
\end{array}\right)D^{3}+\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)D$

  * $(11)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}1&0\\\ 0&b_{2}+u^{2}\\\
\end{array}\right)D+\left(\begin{array}[]{cc}0&0\\\ 0&u^{2}_{x}\\\
\end{array}\right)$

  * $(12)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}-1&0\\\ 0&b_{2}+u^{2}\\\
\end{array}\right)D+\left(\begin{array}[]{cc}0&0\\\ 0&u^{2}_{x}\\\
\end{array}\right)$

  * $(21)$:

$\left(\begin{array}[]{cc}0&h_{1}\\\ h_{1}&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}0&1\\\ 1&u^{1}\\\
\end{array}\right)D+\left(\begin{array}[]{cc}0&0\\\ 0&u^{1}_{x}\\\
\end{array}\right)$

  * $(31):$

$\left(\begin{array}[]{cc}0&h_{1}\\\ h_{1}&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}0&1+u^{1}\\\
1+u^{1}&b_{2}+u^{2}\\\
\end{array}\right)D+\left(\begin{array}[]{cc}0&u^{1}_{x}\\\
u^{1}_{x}&u^{2}_{x}\\\ \end{array}\right)$

  * $(41)$:

$\left(\begin{array}[]{cc}h_{1}&0\\\ 0&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}b_{1}+u^{1}&0\\\
0&b_{2}+u^{2}\\\ \end{array}\right)D+\left(\begin{array}[]{cc}u^{1}_{x}&0\\\
0&u^{2}_{x}\\\ \end{array}\right)$

  * $(51)$:

$\left(\begin{array}[]{cc}-h_{2}&h_{1}\\\ h_{1}&h_{2}\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cc}-b_{2}-u^{2}&b_{1}+u^{1}\\\
b_{1}+u^{1}&b_{2}+u^{2}\\\
\end{array}\right)D+\left(\begin{array}[]{cc}-u^{2}_{x}&u^{1}_{x}\\\
u^{1}_{x}&u^{2}_{x}\\\ \end{array}\right)$

The families of normal forms in Theorem 2.3 contain brackets of many famous
two-component integrable systems: Hirota-Satsuma equation (family $41$),
Antonowitcz-Fordy dispersive water equation (family $31$), Ito system (family
$41$) just to name the few.

Now let us go back to the arbitrary dimension $n$. The Hamiltonian operators
$\mathcal{A}_{g}$ and $\mathcal{B}_{h}$ are the same as in the statement of
the Theorem 2.2. We call the sequence of functionals $\mathcal{H}_{i},i\geq 1$
the multicomponent Harry Dym hierarchy if it satisfies the conditions

$\displaystyle u^{\alpha}_{t_{0}}$ $\displaystyle=\mathcal{A}^{\alpha
q}_{g}\frac{\delta\mathcal{H}_{1}}{\delta u^{q}}=0,$ (25) $\displaystyle
u^{\alpha}_{t_{i}}$ $\displaystyle=\mathcal{A}^{\alpha
q}_{g}\frac{\delta\mathcal{H}_{i+1}}{\delta u^{q}}=\mathcal{B}^{\alpha
q}_{h}\frac{\delta\mathcal{H}_{i}}{\delta u^{q}},\quad i\geq 2.$

The equation $u^{\alpha}_{t_{1}}$ we call multicomponent Harry Dym equation.

We say, that matrix $L$ is a good square root of $R$ if there exists a
polynomial $p(t)$ (may be with complex coefficients), such that $p(R)=L$ and
$L^{2}=M$. If $R$ is non-degenerate, then at least one good root always exists
(Lemma 6.2).

The next theorem shows, that in coordinates, described in Theorem 2.2, the
densities for the multicomponent Harry Dym hierarchy 25 can be found by purely
algebraic means, without any integration.

######  Theorem 2.4.

Let $a^{\alpha\beta}_{s},b^{\alpha\beta},h^{\alpha\beta}$ be a Frobenius
triple, written in a given basis. Define $r^{\beta}_{p}=\frac{1}{2}h^{\beta
q}b_{qp}$ and $c^{\beta}_{ps}=\frac{1}{2}b_{pq}a^{q\beta}_{s}$, where $b_{pq}$
are components of the inverse of $b^{\alpha\beta}$. Denote $l^{\beta}_{s}$ to
be the components of a good square root of matrix $r^{\beta}_{p}$.

Consider a system of algebraic equations

$u^{\beta}=\mathrm{v}^{\beta}+\frac{1}{2}c^{\beta}_{ps}\mathrm{v}^{p}\mathrm{v}^{s}+l^{\beta}_{s}\mathrm{v}^{s}_{x},\quad\beta=1,\dots,n.$
(26)

The solution to this system is a collection of series
$\mathrm{v}^{1},\dots,\mathrm{v}^{n}$ in the form
$\mathrm{v}^{\beta}=\mathcal{v}^{\beta}_{0}+\mathcal{v}^{\beta}_{1}+\mathcal{v}^{\beta}_{2}+\dots$.
Here $\mathcal{v}^{\beta}_{i}$ are either zero or a homogeneous differential
polynomial of degree $i-1$. Take

$\mathcal{H}^{\alpha}_{i}=\int\limits^{2\pi}_{0}v^{\alpha}_{2i-1}\mathrm{d}x,\quad
i=1,\dots.$ (27)

Then the following holds.

  1. 1.

For each $\alpha=1,\dots,n$ the series $\mathcal{H}^{\alpha}_{i}$ defines a
multicomponent Harry Dym hierarchy

$\displaystyle u^{\beta}_{t_{0}^{\alpha}}$ $\displaystyle=\mathcal{A}^{\beta
q}_{g}\frac{\delta\mathcal{H}^{\alpha}_{1}}{\delta u^{q}}=0,$ (28)
$\displaystyle u^{\beta}_{t_{i}^{\alpha}}$ $\displaystyle=\mathcal{A}^{\beta
q}_{g}\frac{\delta\mathcal{H}^{\alpha}_{i+1}}{\delta
u^{q}}=\mathcal{B}_{h}^{\beta q}\frac{\delta\mathcal{H}^{\alpha}_{i+1}}{\delta
u^{q}},\quad i\geq 2.$

for the Hamiltonian operators $\mathcal{A}^{\beta q}_{g}=2(b^{\beta
q}+a^{\beta q}_{s}u^{s})D+a^{\beta q}_{s}u^{s}_{x}$ and
$\mathcal{B}_{h}^{\beta q}=h^{\beta q}D^{3}$

  2. 2.

For arbitrary $\alpha,\beta$ and $i,j$ the flows $u_{t^{\alpha}_{i}}$ and
$u_{t^{\beta}_{j}}$ commute. The functional $\mathcal{H}^{\alpha}_{i}$ is a
conservation law of $u_{t^{\beta}_{j}}$

  3. 3.

Consider the sequence of functionals $\mathcal{H}_{i},i\geq 1$, which define
the multicompomemt Harry Dym hierarchy with respect to the the Hamiltonian
operators $\mathcal{A}^{\beta q}_{g}=2(b^{\beta q}+a^{\beta
q}_{s}u^{s})D+a^{\beta q}_{s}u^{s}_{x}$ and $\mathcal{B}_{h}^{\beta
q}=h^{\beta q}D^{3}$. Assume in addition, that each $\mathcal{H}_{i}$
possesses the following property: it is either zero or a homogeneous
differential polynomial of degree $2i-1$. Then there exists a collection of
constants $c_{0},c_{1},\dots,c_{n}$, such that

$\displaystyle\mathcal{H}_{1}$
$\displaystyle=c_{0}+c_{1}\mathcal{H}^{1}_{1}+\dots+c_{n}\mathcal{H}^{n}_{1},$
$\displaystyle\mathcal{H}_{i}$
$\displaystyle=c_{1}\mathcal{H}^{1}_{i}+\dots+c_{n}\mathcal{H}^{n}_{i},\quad
i>1.$

In particular, the linear space of the multicomponent Harry Dym hierarchies
with the property given above is finite-dimensional and sequences
$\mathcal{H}^{\alpha}_{i},i\geq 1,\alpha=1,\dots,n$ together with constant
functional form a basis in this space.

  4. 4.

If in addition $a^{\alpha\beta}_{s}$ has unity, then one can replace equation
(26) with simpler Miura type equation

$u^{\beta}=\frac{1}{2}c^{\beta}_{ps}\mathrm{v}^{p}\mathrm{v}^{s}+l^{\beta}_{s}\mathrm{v}^{s}_{x},\quad\beta=1,\dots,n.$
(29)

and Hamiltonian operator of order one in formula (28) with $\mathcal{A}^{\beta
q}_{g}=2a^{\beta q}_{s}u^{s}D+a^{\beta q}_{s}u^{s}_{x}$.

Assume now that in Frobenius triple
$a^{\alpha\beta}_{s},b^{\alpha\beta},h^{\alpha\beta}$ the commutative
associative algebra has unity $e=(e_{1},\dots,e_{n})$. For Hamiltonian
operators $\mathcal{A}^{\beta q}_{g}=2a^{\beta q}_{s}u^{s}D+a^{\beta
q}_{s}u^{s}_{x}$ and $\mathcal{B}_{h}^{\beta q}=h^{\beta q}D^{3}$ (these are
the Hamiltonian operators from the fourth statement of the Theorem 2.4) we
write negative hierarchy for $i=0,-1,-2$ we get:

$\displaystyle u^{\beta}_{t_{0}}$ $\displaystyle=h^{\beta
q}D^{3}\Big{(}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta u^{q}}\Big{)}=0,$ (30)
$\displaystyle u^{\beta}_{t_{-1}}$ $\displaystyle=2a^{\beta
q}_{s}u^{s}D\Big{(}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta
u^{q}}\Big{)}+a^{\beta
q}_{s}u^{s}_{x}\frac{\delta\bar{\mathcal{H}}_{0}}{\delta u^{q}}=h^{\beta
q}D^{3}\Big{(}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u^{q}}\Big{)},$
$\displaystyle u^{\beta}_{t_{-2}}$ $\displaystyle=2a^{\beta
q}_{s}u^{s}D\Big{(}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta
u^{q}}\Big{)}+a^{\beta
q}_{s}u^{s}_{x}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u^{q}}$

Consider functional
$\bar{\mathcal{H}}_{0}=\int^{2\pi}_{0}e_{\alpha}u^{\alpha}\mathrm{d}x$. The
equation for $i=-1$ takes form

$u^{\alpha}_{t_{-1}}=u^{\alpha}_{x}=h^{\alpha
q}D^{3}\Big{(}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u^{q}}\Big{)}.$

We introduce new variables $v^{\alpha}$ as

$v^{p}_{xxx}=\mathcal{B}^{pq}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta
u^{q}}.$

In given coordinates this yields
$h^{qp}\frac{\delta\bar{\mathcal{H}}_{-1}}{\delta u^{q}}=v^{p}$. We get, that
$u^{\alpha}=v^{\alpha}_{xx}$. The equation (30) for $i=-2$ takes form

$v^{\alpha}_{xxt_{-2}}=2a^{\alpha q}_{s}v^{s}_{xx}h_{qp}v^{p}_{x}+a^{\alpha
q}_{s}v^{s}_{xxx}h_{qp}v^{p}.$

Denote $\bar{c}^{\alpha}_{ps}=h_{pq}a^{\alpha q}_{s}$. Renaming $t_{-2}$ as
$t$, we integrate both sides of the equation, using property
$\bar{c}^{\alpha}_{ps}=\bar{c}^{\alpha}_{sp}$ (Lemma 6.1). Taking the
integration constants we arrive to the system

$\displaystyle
v^{\alpha}_{xt}=\frac{1}{2}\bar{c}^{\alpha}_{ps}\big{(}v^{p}_{xx}v^{s}+v^{p}_{x}v^{s}_{x}+v^{p}v^{s}_{xx}\big{)}.$
(31)

We call this system the multicomponent Hunter-Saxton equation.

By construction we see, that these equations are defined for arbitrary
commutative associative algebra. The existence of Frobenius form is necessary
and sufficient for this equation to be bihamiltonian with respect to
$\mathcal{A}_{g}$ and $\mathcal{B}_{g}$. The conservation laws for this
equation are obtained from Theorem 2.4.

In the end of the paper we consider two examples.

###  2.1 Example I: Multicomponent Harry Dym and Hunter-Saxton equations,
associated with algebra $\mathfrak{t}_{n}$

Following [22] we define commutative associative algebra with unity
$\mathfrak{t}_{n}$: in given basis $\eta^{1},\dots,\eta^{n}$ the
multiplication is defined as

$\eta^{i}\star\eta^{j}=\delta^{i+j-1}_{k}\eta^{k}.$ (32)

We have that $\eta^{1}$ is unity and for $k\geq 2$

$\eta^{k}=\underbrace{\eta^{2}\star\dots\star\eta^{2}}_{\text{$k-1$ times}}.$

All the calculations in Theorem 2.4 for algebra $\mathfrak{t}_{n}$ can be done
explicitly.

Assume that we have different basis $\bar{\eta}^{i}$, such that in dual basis
$\eta_{i}=\frac{1}{2}b_{iq}\bar{\eta}^{q}$ the structure constants
$c^{\beta}_{ps}$ are in the form (32). Matrix $U$ is defined as
$U\eta=\eta\star u$, where $u=\eta_{1}u^{1}+\dots+\eta_{n}u^{n}$. In given
coordinates

$U=\left(\begin{array}[]{cccccc}u^{1}&0&0&\dots&0&0\\\
u^{2}&u^{1}&0&\dots&0&0\\\ u^{3}&u^{2}&u^{1}&\dots&0&0\\\ &&&\ddots&&\\\
u^{n-1}&u^{n-2}&u^{n-3}&\dots&u^{1}&0\\\
u^{n}&u^{n-1}&u^{n-2}&\dots&u^{2}&u^{1}\end{array}\right).$ (33)

The second matrix we need for calculation is

$\mathtt{V}=\left(\begin{array}[]{cccccc}\mathrm{v}^{1}&0&0&\dots&0&0\\\
\mathrm{v}^{2}&\mathrm{v}^{1}&0&\dots&0&0\\\
\mathrm{v}^{3}&\mathrm{v}^{2}&\mathrm{v}^{1}&\dots&0&0\\\ &&&\ddots&&\\\
\mathrm{v}^{n-1}&\mathrm{v}^{n-2}&\mathrm{v}^{n-3}&\dots&\mathrm{v}^{1}&0\\\
\mathrm{v}^{n}&\mathrm{v}^{n-1}&\mathrm{v}^{n-2}&\dots&\mathrm{v}^{2}&\mathrm{v}^{1}\end{array}\right).$
(34)

The components of this matrix are series
$\mathrm{v}^{\alpha}=\mathcal{v}^{1}+\mathcal{v}^{2}+\dots$ (see Theorem 2.4).
The matrix $\mathtt{V}$ naturally decomposes into a series
$\mathtt{V}=V_{1}+V_{2}+\dots$, where entries of $V_{i}$ are either zero or
$\mathcal{v}^{\alpha}_{i}$.

In terms of $U$ and $\mathtt{V}$ the equation (29) can be written as

$U=\frac{1}{2}\mathtt{V}^{2}+\mathtt{V}_{x},$ (35)

where the terms of matrix $\mathtt{V}_{x}$ are the derivative in $x$ of the
corresponding terms of $\mathtt{V}$. This system looks exactly like the system
(5) in one-dimensional case with scalars being replaced by matrices. The
equation (35) produces a system of algebraic equations on $V_{i}$

$\displaystyle U$ $\displaystyle=\frac{1}{2}V_{1}^{2},$ (36) $\displaystyle
V_{i+1}$
$\displaystyle=-V_{1}^{-1}\Bigg{(}\,\frac{1}{2}\sum\limits_{j=2}^{i}V_{j}V_{i+2-j}+(V_{i})_{x}\Bigg{)},\quad
i\geq 1.$

To solve the first equation in (36) recall, that

$(1+t)^{\alpha}=\sum\limits_{i=0}^{2\pi}\binom{\alpha}{i}t^{i},\quad\binom{\alpha}{i}=\frac{\alpha(\alpha-1)\dots(\alpha-i+1)}{i!}.$
(37)

Using this notation we get

$V_{1}=\sqrt{2u^{1}}\Big{(}\operatorname{Id}+\frac{1}{u_{1}}(U-u_{1}\operatorname{Id})\Big{)}^{\frac{1}{2}}=\sqrt{2u^{1}}\Bigg{(}\sum\limits_{i=0}^{n-1}\binom{\frac{1}{2}}{i}\Big{(}\frac{1}{u^{1}}U-\operatorname{Id}\Big{)}^{i}\Bigg{)}.$
(38)

Here we used that $U-u_{1}\operatorname{Id}$ is nilpotent. The system 36 takes
form

$\displaystyle V_{1}$
$\displaystyle=\sqrt{2u^{1}}\Bigg{(}\sum\limits_{i=0}^{n-1}\binom{\frac{1}{2}}{i}\Big{(}\frac{1}{u^{1}}U-\operatorname{Id}\Big{)}^{i}\Bigg{)},$
(39) $\displaystyle V_{i+1}$
$\displaystyle=-\frac{1}{\sqrt{2u^{1}}}\Bigg{(}\sum\limits_{i=0}^{n-1}\binom{-\frac{1}{2}}{i}\Big{(}\frac{1}{u^{1}}U-\operatorname{Id}\Big{)}^{i}\Bigg{)}\Bigg{(}\,\frac{1}{2}\sum\limits_{j=2}^{i}V_{j}V_{i+2-j}+(V_{i})_{x}\Bigg{)},\quad
i\geq 1.$

In case of $n=2$ the formula (38) produces

$\mathcal{H}^{1}_{1}=\int^{2\pi}_{0}\sqrt{2u^{1}}\mathrm{d}x,\quad\mathcal{H}^{2}_{1}=\int^{2\pi}_{0}\frac{u^{2}}{\sqrt{2u^{1}}}\mathrm{d}x.$

These are the Hamiltonians of the coupled Harry Dym equation, obtained by
Antonowicz and Fordy in [23]. For $n=3$ the formula (38) produces

$\mathcal{H}^{1}_{1}=\int^{2\pi}_{0}\sqrt{2u^{1}}\mathrm{d}x,\quad\mathcal{H}^{2}_{1}=\int^{2\pi}_{0}\frac{u^{2}}{\sqrt{2u^{1}}}\mathrm{d}x,\quad\mathcal{H}^{3}_{1}=\int^{2\pi}_{0}\Bigg{(}\frac{u^{3}}{(2u^{1})^{\frac{1}{2}}}-\frac{(u^{2})^{2}}{(2u^{1})^{\frac{3}{2}}}\Bigg{)}\mathrm{d}x.$

Note that $\mathcal{A}_{g}$ in the fourth statement of Theorem 2.4 does not
depend on $b^{\alpha\beta}$. Thus, we may assume that
$2b^{\alpha\beta}=h^{\alpha\beta}$. In particular, the constants
$\bar{c}^{\alpha}_{ps}$, that appear in construction of multicomponent Hunter-
Saxton equation, coincide with $c^{\alpha}_{ps}$. We introduce matrix

$V=\left(\begin{array}[]{cccccc}v^{1}&0&0&\dots&0&0\\\
v^{2}&v^{1}&0&\dots&0&0\\\ v^{3}&v^{2}&u^{1}&\dots&0&0\\\ &&&\ddots&&\\\
v^{n-1}&v^{n-2}&v^{n-3}&\dots&v^{1}&0\\\
v^{n}&v^{n-1}&v^{n-2}&\dots&v^{2}&v^{1}\end{array}\right).$ (40)

Basically matrix (40) is matrix (33) with $u^{\alpha}$ replaced by
$v^{\alpha}$. By construction of coordinates $v^{i}$ we have $V_{xx}=U$. The
multicomponent Hunter-Saxton equation can be written as

$V_{xt}=V_{xx}V+\frac{1}{2}V^{2}_{x}.$ (41)

The system (41) seems to be new. For $n=2$ we get

$\displaystyle v^{1}_{xt}$
$\displaystyle=v^{1}_{xx}v^{1}+\frac{1}{2}(v^{1}_{x})^{2},$ $\displaystyle
v^{2}_{xt}$
$\displaystyle=v^{1}_{xx}v^{2}+v^{1}_{x}v^{2}_{x}+v^{1}v^{2}_{xx}.$

One notices the differences between this multicomponent version of Hunter-
Saxton equation and the so-called 2-HS equation. The latter was obtained as a
short-wave limit of 2-CH equation (see [26] for formulas and further
references on the subject and its applications). For $n=3$ the multicomponent
Hunter-Saxton equation in coordinates $v^{1},v^{2},v^{3}$ takes form

$\displaystyle\begin{aligned}
v^{1}_{xt}&=v^{1}_{xx}v^{1}+\frac{1}{2}(v^{1}_{x})^{2},\\\
v^{2}_{xt}&=v^{1}_{xx}v^{2}+v^{1}_{x}v^{2}_{x}+v^{2}v^{1}_{xx},\\\
v^{3}_{xt}&=v^{2}_{xx}v^{2}+\frac{1}{2}(v^{2}_{x})^{2}+v^{1}v^{3}_{xx}+v^{1}_{x}v^{3}_{x}+v^{1}_{xx}v^{3}.\end{aligned}$

###  2.2 Example II: The Hamiltonian operators, multicomonent Harry Dym and
Hunter-Saxton equations, associated with certain commutative associative
algebra in dimension $4$

Consider a commutative associative algebra with unity and Frobenius form.
Proposition 7.3 from Appendix implies, that every such an algebra is
decomposed into the direct sum of irreducible terms. For dimensions two and
three the only terms, that appear in such a decomposition are
$\mathfrak{t}_{1},\mathfrak{t}_{2},\mathfrak{t}_{3}$ and
$\mathfrak{t}_{1}^{\mathbb{C}}$ ($\mathfrak{t}_{1}^{\mathbb{C}}$ stands for
the real form of complex one-dimensional algebra $\mathfrak{t}_{1}$).

The first commutative associative algebra that differs from the aforementioned
series of examples, appears in dimension $4$. In given basis
$\eta^{1},\dots,\eta^{4}$ define the non-zero structure relations as

$\displaystyle\eta^{1}\star\eta^{i}$ $\displaystyle=\eta^{i},\quad
i=1,\dots,4,$ $\displaystyle\eta^{2}\star\eta^{2}$
$\displaystyle=\eta^{4},\quad\eta^{3}\star\eta^{3}=\eta^{4}.$

To simplify the calculations, we take $h^{1}=1,h_{i}=0,i=2,3,4$ and define the
Frobenius forms as
$2b^{\alpha\beta}=h^{\alpha\beta}=a^{\alpha\beta}_{s}h^{s}$.

The Hamiltonian operator of the inhomogeneous Poisson bracket of type $1+3$,
associated with this Frobenius triple in coordinates, defined by Theorem 2.2,
is

$\mathcal{P}^{\alpha\beta}=\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&1&0&0\\\
0&0&1&0\\\ 1&0&0&0\\\
\end{array}\right)D^{3}+2\left(\begin{array}[]{cccc}u^{1}&u^{2}&u^{3}&u^{4}\\\
u^{2}&u^{4}&0&0\\\ u^{3}&0&u^{4}&0\\\ u^{4}&0&0&0\\\
\end{array}\right)D+\left(\begin{array}[]{cccc}u_{x}^{1}&u_{x}^{2}&u_{x}^{3}&u_{x}^{4}\\\
u_{x}^{2}&u_{x}^{4}&0&0\\\ u_{x}^{3}&0&u_{x}^{4}&0\\\ u_{x}^{4}&0&0&0\\\
\end{array}\right).$

Lowering the indices $\eta_{i}=h_{iq}\eta^{i}$ we obtain a commutative
associative structure on tangent space

$\displaystyle\eta_{4}\star\eta_{i}$ $\displaystyle=\eta_{i},\quad
i=1,\dots,4,$ $\displaystyle\eta_{2}\star\eta_{2}$
$\displaystyle=\eta_{1},\quad\eta_{3}\star\eta_{3}=\eta_{1}.$

We apply Theorem 2.4 to find the Casimirs of this bracket. To do that we take
the equation (29) with $l^{\beta}_{p}=\delta^{\beta}_{p}$ takes form

$\displaystyle u^{1}$
$\displaystyle=\mathrm{v}^{1}\mathrm{v}^{4}+\frac{1}{2}(\mathrm{v}^{2})^{2}+\frac{1}{2}(\mathrm{v}^{3})^{2}+\mathrm{v}^{1}_{x},$
(42) $\displaystyle u^{2}$
$\displaystyle=\mathrm{v}^{4}\mathrm{v}^{2}+\mathrm{v}^{2}_{x},$
$\displaystyle u^{3}$
$\displaystyle=\mathrm{v}^{4}\mathrm{v}^{3}+\mathrm{v}^{3}_{x},$
$\displaystyle u^{4}$
$\displaystyle=\frac{1}{2}(\mathrm{v}^{4})^{2}+\mathrm{v}^{4}_{x}.$

We get the following collection of series of functionals
$\mathcal{H}^{\alpha}_{1}$:

$\mathcal{H}^{4}_{1}=\int\limits^{2\pi}_{0}\sqrt{2u^{4}}\mathrm{d}x,\quad\mathcal{H}^{3}_{1}=\int\limits^{2\pi}_{0}\frac{u^{3}}{\sqrt{2u^{4}}}\mathrm{d}x,\quad\mathcal{H}^{2}_{1}=\int\limits^{2\pi}_{0}\frac{u^{2}}{\sqrt{2u^{4}}}\mathrm{d}x,\quad\mathcal{H}^{1}_{1}=\int\limits^{2\pi}_{0}\Bigg{(}\frac{u^{1}}{(2u^{4})^{\frac{1}{2}}}-\frac{1}{2}\frac{(u^{2})^{2}+(u^{3})^{2}}{(2u^{4})^{\frac{3}{2}}}\Bigg{)}\mathrm{d}x.$

For $\alpha=1,i=1$ the bihamiltonian flow (28) of Theorem 2.4 is written as

$\displaystyle u^{1}_{t}$
$\displaystyle=\Bigg{(}\frac{3}{2}\frac{(u^{2})^{2}+(u^{3})^{2}}{(2u^{4})^{\frac{5}{2}}}-\frac{u^{1}}{(2u^{4})^{\frac{3}{2}}}\Bigg{)}_{xxx},$
(43) $\displaystyle u^{2}_{t}$
$\displaystyle=-\Bigg{(}\frac{u^{2}}{(2u^{4})^{\frac{3}{2}}}\Bigg{)}_{xxx},$
$\displaystyle u^{3}_{t}$
$\displaystyle=-\Bigg{(}\frac{u^{3}}{(2u^{4})^{\frac{3}{2}}}\Bigg{)}_{xxx},$
$\displaystyle u^{4}_{t}$
$\displaystyle=\Bigg{(}\frac{1}{\sqrt{2u^{4}}}\Bigg{)}_{xxx}.$

Here we renamed $t^{1}_{1}$ as $t$. The equation (43) is the four-component
Harry Dym equation. The corresponding 4-component Hunter-Saxton equations is
written as

$\displaystyle v^{1}_{xt}$
$\displaystyle=v^{1}_{xx}v^{4}+v^{1}_{x}v^{4}_{x}+v^{1}v^{4}_{xx}+v^{2}_{xx}v^{2}+\frac{1}{2}(v^{2}_{x})^{2}+v^{3}_{xx}v^{3}+\frac{1}{2}(v^{3}_{x})^{2},$
(44) $\displaystyle v^{2}_{xt}$
$\displaystyle=v^{4}_{xx}v^{2}+v^{4}_{x}v^{2}_{x}+v^{4}v^{2}_{xx},$
$\displaystyle v^{3}_{xt}$
$\displaystyle=v^{4}_{xx}v^{3}+v^{4}_{x}v^{3}_{x}+v^{4}v^{3}_{xx},$
$\displaystyle v^{4}_{xt}$
$\displaystyle=v^{4}_{xx}v^{4}+\frac{1}{2}(v^{4})^{2}$

This system looks like two entangled two-component Harry Dym systems from the
previous example.

##  3 Proof of Theorem 2.1

Assume that $\mathcal{P}^{\alpha\beta}$ is written in Darboux coordinates
$u^{1},\dots,u^{n}$. Denote $\Gamma^{\beta}_{rs}$ to be the Christoffel
symbols of the flat symmetric connection, for which these coordinates are
flat.

Consider another coordinate system $\bar{u}^{1},\dots,\bar{u}^{n}$ and denote
the Christoffel symbols of the same connection in these coordinates as
$\bar{\Gamma}^{\beta}_{rs}$. We have

$\displaystyle 0=D(\delta^{\beta}_{r})$
$\displaystyle=D\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\frac{\partial
u^{p}}{\partial\bar{u}^{r}}\Big{)}=D\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}\frac{\partial
u^{p}}{\partial\bar{u}^{r}}+\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}D\Big{(}\frac{\partial u^{p}}{\partial\bar{u}^{r}}\Big{)}=$
$\displaystyle=D\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}\frac{\partial
u^{p}}{\partial\bar{u}^{r}}+\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\frac{\partial^{2}u^{p}}{\partial\bar{u}^{r}\partial\bar{u}^{s}}\bar{u}^{s}_{x}=D\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}\frac{\partial
u^{p}}{\partial\bar{u}^{r}}+\bar{\Gamma}^{\beta}_{rs}\bar{u}^{s}_{x}.$

Here we have used the standard transformation rule for the Christoffel
symbols. After multiplying by Jacobi matrix $\frac{\partial\bar{u}}{\partial
u}$ both sides and rearranging the terms, we get formula

$D\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}=-\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\bar{\Gamma}^{\beta}_{qs}\bar{u}^{s}_{x}=\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\tau^{\beta}_{q}(1).$ (45)

Let us show that the more general formula holds

$D^{j}\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}=\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\tau^{\beta}_{q}(j),\quad j\geq 1.$ (46)

We prove it by induction. The basis is provided by formula (45). The induction
step is

$\displaystyle D^{j+1}\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{p}}\Big{)}$ $\displaystyle=D\Big{(}\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\tau_{q}^{\beta}(j)\Big{)}=D\Big{(}\frac{\partial\bar{u}^{m}}{\partial
u^{q}}\Big{)}\tau^{\beta}_{m}(j)+\frac{\partial\bar{u}^{q}}{\partial
u^{p}}D\big{(}\tau^{\beta}_{q}(j)\big{)}=$
$\displaystyle=\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\tau^{m}_{q}(1)\tau^{\beta}_{m}(j)+\frac{\partial\bar{u}^{q}}{\partial
u^{p}}D\big{(}\tau^{\beta}_{q}(j)\big{)}=\frac{\partial\bar{u}^{q}}{\partial
u^{p}}\tau^{\beta}_{q}(j+1).$

The formula (46) is proved.

Now consider a pair of functionals
$\mathcal{H}_{1}=\int_{0}^{2\pi}\mathcal{h}_{1}\mathrm{d}x$ and
$\mathcal{H}_{2}=\int_{0}^{2\pi}\mathcal{h}_{2}\mathrm{d}x$. By direct
computation we get

$\displaystyle\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}$
$\displaystyle=\int\limits^{2\pi}_{0}h^{pq}\frac{\delta\mathcal{h}_{1}}{\delta
u^{p}}D^{k}\Big{(}\frac{\delta\mathcal{h}_{2}}{\delta
u^{q}}\Big{)}\mathrm{d}x=\int\limits^{2\pi}_{0}h^{pq}\frac{\partial\bar{u}^{\alpha}}{\partial
u^{p}}\frac{\delta\mathcal{h}_{1}}{\delta\bar{u}^{\alpha}}D^{k}\Big{(}\frac{\delta\mathcal{h}_{2}}{\delta\bar{u}^{\beta}}\frac{\partial\bar{u}^{\beta}}{\partial
u^{q}}\Big{)}\mathrm{d}x=$ (47)
$\displaystyle=\int\limits^{2\pi}_{0}h^{pq}\frac{\partial\bar{u}^{\alpha}}{\partial
u^{p}}\frac{\delta\mathcal{h}_{1}}{\delta\bar{u}^{\alpha}}\Bigg{(}\sum\limits_{j=0}^{k}\binom{j}{k}D^{j}\Big{(}\frac{\partial\bar{u}^{\beta}}{\partial
u^{q}}\Big{)}D^{k-j}\Big{(}\frac{\delta\mathcal{h}_{2}}{\delta\bar{u}^{\beta}}\Big{)}\Bigg{)}\mathrm{d}x=$
$\displaystyle=\int\limits^{2\pi}_{0}\Bigg{(}\bar{h}^{\alpha\beta}\frac{\delta\mathcal{h}_{1}}{\delta\bar{u}^{\alpha}}\frac{\delta\mathcal{h}_{2}}{\delta\bar{u}^{\beta}}+\sum\limits_{j=1}^{k}\binom{j}{k}h^{\alpha
q}\tau^{\beta}_{q}(j)\frac{\delta\mathcal{h}_{1}}{\delta\bar{u}^{\alpha}}D^{k-j}\Big{(}\frac{\delta\mathcal{h}_{2}}{\delta\bar{u}^{\beta}}\Big{)}\Bigg{)}\mathrm{d}x=$
$\displaystyle=\int\limits^{2\pi}_{0}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}h^{\alpha\beta}D^{k}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Big{)}\mathrm{d}x+\int\limits^{\infty}_{0}\frac{\delta\mathcal{H}_{1}}{\delta
u^{\alpha}}\sum\limits_{j=1}^{k}\mathcal{P}^{\alpha\beta}_{\mathrm{j}}D^{k-j}\Big{(}\frac{\delta\mathcal{H}_{2}}{\delta
u^{\beta}}\Big{)}\mathrm{d}x$

Here we applied formula (46).

Recall, that $\binom{1}{k}=k$. This implies, that
$\mathcal{P}_{1}^{\alpha\beta}=-kh^{\alpha q}\Gamma^{\beta}_{qs}u^{s}_{x}$.
Thus, the first statement of the Theorem 2.1 holds.

The coordinates $u^{1},\dots,u^{n}$ are the Darboux coordinates for Poisson
bracket and flat coordinates for $\Gamma^{\beta}_{qs}$. By construction

$0=\frac{\partial h^{\alpha\beta}}{\partial u^{s}}=\nabla_{s}h^{\alpha\beta},$

thus, the second statement of Theorem 2.1 is true. The third statement of the
Theorem follows directly from (46).

Finally, assume that for a given odd $k$ (for even the proof is similar) we
have a non-degenerate symmetric tensor $h^{\alpha\beta}$ of type $(2,0)$ and
flat symmetric connection $\Gamma^{\beta}_{qs}$ with condition $\nabla
h^{\alpha\beta}=0$. Taking $u^{1},\dots,u^{n}$ to be the flat coordinates of
the connection, we arrive to the situation we have started the proof with.
Thus, the theorem is proved.

##  4 Proof of Theorem 2.2 and Corollary 2.1

First, let us show, that the conditions of Theorem 2.2 are necessary.

Fix Darboux coordinates $u^{1},\dots,u^{n}$ of $\mathcal{B}_{h}$. The
Christoffel symbols of the Levi-Civita connection of metric $g$ in these
coordinates are denoted as $\Gamma^{\beta}_{qs}$. The contrvariant Christoffel
symbols are defined as (our definition differs by the sign from the one in
[24]) $\Gamma^{\alpha\beta}_{s}=g^{\alpha q}\Gamma^{\beta}_{qs}$. The
condition $\nabla_{q}g^{\alpha\beta}=0$ yields

$\frac{\partial g^{\alpha\beta}}{\partial
u^{q}}+\Gamma^{\alpha\beta}_{q}+\Gamma^{\beta\alpha}_{q}=0.$ (48)

The symmetry of Levi-Civita connection yields the second condition

$\Gamma^{\alpha\beta}_{q}g^{q\gamma}=\Gamma^{\gamma\beta}_{q}g^{q\alpha}.$
(49)

Consider the functionals

$\mathcal{H}_{1}=\int\limits^{2\pi}_{0}\mathcal{a}_{\alpha}u^{\alpha}\mathrm{d}x,\quad\mathcal{H}_{2}=\int\limits^{2\pi}_{0}\mathcal{b}_{\beta}u^{\beta}\mathrm{d}x,\quad\mathcal{H}_{3}=\int\limits^{2\pi}_{0}\mathcal{c}_{\gamma}u^{\gamma}\mathrm{d}x$

where $\mathcal{a}_{\alpha},\mathcal{b}_{\beta},\mathcal{c}_{\gamma}$ for
$\alpha,\beta,\gamma=1,\dots,n$ are arbitrary $3n$ functions of variable $x$.
We introduce notation $\mathcal{a}^{(i)}_{\alpha}=D^{i}(\mathcal{a}_{\alpha})$
and same for $\mathcal{b}_{\beta},\mathcal{c}_{\gamma}$. In these notations
the compatibility condition for corresponding brackets is written as follows:
for arbitrary functionals $\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}_{3}$
one has

$\displaystyle
0=\\{\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{g},\mathcal{H}_{3}\\}_{h}+\\{\\{\mathcal{H}_{2},\mathcal{H}_{3}\\}_{g},\mathcal{H}_{1}\\}_{h}+\\{\\{\mathcal{H}_{3},\mathcal{H}_{1}\\}_{g},\mathcal{H}_{2}\\}_{h}+\\{\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}_{h},\mathcal{H}_{3}\\}_{g}+$
(50) $\displaystyle+$
$\displaystyle\\{\\{\mathcal{H}_{2},\mathcal{H}_{3}\\}_{h},\mathcal{H}_{1}\\}_{g}+\\{\\{\mathcal{H}_{3},\mathcal{H}_{1}\\}_{h},\mathcal{H}_{2}\\}_{g}=$
$\displaystyle=$ $\displaystyle\int\limits_{0}^{2\pi}\Bigg{(}\frac{\partial
g^{\alpha\beta}}{\partial
u^{q}}h^{q\gamma}\mathcal{a}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(k)}_{\gamma}-\frac{\partial\Gamma^{\alpha\beta}_{s}}{\partial
u^{q}}h^{q\gamma}u^{s}_{x}\mathcal{a}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}-\Gamma^{\alpha\beta}_{q}h^{q\gamma}\mathcal{a}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k+1)}_{\gamma}\Bigg{)}\mathrm{d}x+\text{c.p}=$
$\displaystyle=$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{(}\Gamma^{\alpha\beta}_{q}h^{q\gamma}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}-\Gamma^{\beta\alpha}_{q}h^{q\gamma}\mathcal{a}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(k)}_{\gamma}+\Big{(}\frac{\partial\Gamma^{\alpha\beta}_{q}}{\partial
u^{s}}h^{q\gamma}-\frac{\partial\Gamma^{\alpha\beta}_{s}}{\partial
u^{q}}h^{q\gamma}\Big{)}u^{s}_{x}\mathcal{a}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}\Bigg{)}\mathrm{d}x+\text{c.p.}$

Here c.p. stands for cyclic permutations in
$\mathcal{a}_{\alpha},\mathcal{b}_{\beta},\mathcal{c}_{\gamma}$ and we used
identity (48). We denote

$A^{\alpha\beta\gamma}=\Gamma^{\alpha\beta}_{q}h^{q\gamma}\quad\text{ and
}\quad
T^{\alpha\beta\gamma}=\Bigg{(}\frac{\partial\Gamma^{\alpha\beta}_{q}}{\partial
u^{s}}h^{q\gamma}-\frac{\partial\Gamma^{\alpha\beta}_{s}}{\partial
u^{q}}h^{q\gamma}\Bigg{)}u^{s}_{x}.$ (51)

The compatibility condition (50) in these notations takes form

$\displaystyle 0=$
$\displaystyle\int\limits^{2\pi}_{0}A^{\alpha\beta\gamma}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\beta\alpha\gamma}\mathcal{a}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}A^{\beta\gamma\alpha}\mathcal{a}^{(k)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-$
(52) $\displaystyle-$
$\displaystyle\int\limits^{2\pi}_{0}A^{\gamma\beta\alpha}\mathcal{a}^{(k)}_{\gamma}\mathcal{b}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}A^{\gamma\alpha\beta}\mathcal{a}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\alpha\gamma\beta}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+$
$\displaystyle+$
$\displaystyle\int\limits^{2\pi}_{0}T^{\alpha\beta\gamma}\mathcal{a}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}T^{\beta\gamma\alpha}\mathcal{a}^{(k)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}T^{\gamma\alpha\beta}\mathcal{a}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x.$

The following identity holds for arbitrary differential polynomials
$\mathcal{h}_{1},\mathcal{h}_{2}$

$0=\int\limits_{0}^{2\pi}D(\mathcal{h}_{1}\mathcal{h}_{2})\mathrm{d}x=\int\limits_{0}^{2\pi}D(\mathcal{h}_{1})\mathcal{h}_{2}\mathrm{d}x+\int\limits_{0}^{2\pi}\mathcal{h}_{1}D(\mathcal{h}_{2})\mathrm{d}x.$

It is usually referred to as ”integration by parts”. Applying this identity to
the terms in (52) we get

$\displaystyle\int\limits^{2\pi}_{0}A^{\alpha\beta\gamma}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x=-\sum\limits_{i+j+r=k}\binom{k}{i,j,r}\int\limits_{0}^{2\pi}D^{i}\big{(}A^{\alpha\beta\gamma}\big{)}\mathcal{a}^{(1+j)}_{\alpha}\mathcal{b}^{(r)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x,$
$\displaystyle\int\limits^{2\pi}_{0}A^{\beta\alpha\gamma}\mathcal{a}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x=-\sum\limits_{i+j+r=k}\binom{k}{i,j,r}\int\limits^{2\pi}_{0}D^{i}\big{(}A^{\beta\alpha\gamma}\big{)}\mathcal{a}^{(j)}_{\alpha}\mathcal{b}^{(r+1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x,$
$\displaystyle\int\limits^{2\pi}_{0}T^{\alpha\beta\gamma}\mathcal{a}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(k)}_{\gamma}\mathrm{d}x=-\int\limits^{2\pi}_{0}\sum\limits_{i+j+r=k}\binom{k}{i,j,r}D^{i}\big{(}T^{\alpha\beta\gamma}\big{)}\mathcal{a}^{(j)}_{\alpha}\mathcal{b}^{(r)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x,$
$\displaystyle\int\limits^{2\pi}_{0}A^{\gamma\beta\alpha}\mathcal{a}^{(k)}_{\gamma}\mathcal{b}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x=-\int\limits^{2\pi}_{0}D(A^{\gamma\beta\alpha})\mathcal{a}^{(k)}_{\gamma}\mathcal{b}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\gamma\beta\alpha}\mathcal{a}^{(k+1)}_{\gamma}\mathcal{b}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\gamma\beta\alpha}\mathcal{a}^{(k)}_{\gamma}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x,$
$\displaystyle\int\limits^{2\pi}_{0}A^{\gamma\alpha\beta}\mathcal{a}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x=-\int\limits^{2\pi}_{0}D(A^{\gamma\alpha\beta})\mathcal{a}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\gamma\alpha\beta}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(k)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}A^{\gamma\alpha\beta}\mathcal{a}_{\alpha}\mathcal{b}^{(k+1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x.$

Here $\binom{n}{p,q,r}=\frac{n!}{p!q!r!}$ are trinomial coefficients. We
substitute these equations into (52). After denoting the coefficients in front
of the monomial
$\mathcal{a}^{(i)}_{\alpha}\mathcal{b}^{(j)}_{\beta}\mathcal{c}_{\gamma}$ as
$M^{\alpha\beta\gamma}_{ij}$, the condition (52) takes form

$0=\int\limits_{0}^{2\pi}\sum\limits_{i,j}M^{\alpha\beta\gamma}_{ij}\mathcal{a}^{(i)}_{\alpha}\mathcal{b}^{(j)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x.$
(53)

Now recall, that
$\mathcal{a}_{\alpha},\mathcal{b}_{\beta},\mathcal{c}_{\gamma}$ are arbitrary
functions in $x$. We perform a rather standard analytical trick: fix
$x_{0}\in\mathbb{R}$, $\alpha,\beta,\gamma\leq n$ and $i,j$ with $i+j\leq
k+1$. We take $\mathcal{a}_{q}=0$ for $q\neq\alpha$ and $\mathcal{b}_{r}=0$
for $r\neq\beta$. On $\mathcal{a}_{\alpha}$ and $\mathcal{b}_{\beta}$ we put
conditions $\mathcal{a}^{(p)}_{\alpha}(x_{0})=\delta^{p}_{i}$ and
$\mathcal{b}^{(q)}_{\alpha}(x_{0})=\delta^{q}_{j}$.

We take $\mathcal{c}_{q}=0$ for $q\neq\gamma$ and consider the sequence of
smooth functions $\mathcal{c}^{n}_{\gamma}(x)$ which converge to delta-
function $\delta(x-x_{0})$ for $n\to\infty$. We get, that (53) converges to
$M^{\alpha\beta\gamma}_{ij}(x_{0})=0$. By the construction it holds for all
functions $u^{\alpha}$. As $x_{0}$ was chosen arbitrary, we get, that (49) is
equivalent to the collection of the conditions

$M^{\alpha\beta\gamma}_{ij}=0,\quad 1\leq\alpha,\beta,\gamma\leq n,\quad 0\leq
i+j\leq k+1.$ (54)

Now we are going to study this system.

######  Lemma 4.1.

The compatibility condition (52) implies:

  1. 1.

$T^{\alpha\beta\gamma}\equiv 0$

  2. 2.

$A^{\alpha\beta\gamma}$ are constants and totally symmetric in
$\alpha,\beta,\gamma$

###### Proof.

Fix arbitrary $\alpha,\beta,\gamma$. From (54) and (52) we get

$M_{00}^{\alpha\beta\gamma}=-D^{k}\big{(}T^{\alpha\beta\gamma}\big{)}=0.$

Recall that $T^{\alpha\beta\gamma}$ is a homogeneous differential polynomial
of degree one, with coefficients that do not depend on $x$. By Lemma 6.4 the
vanishing of the derivative implies that this polynomial is identically zero.

Now consider the following coefficients:

$\displaystyle
0=M^{\alpha\beta\gamma}_{k+1\,0}=A^{\gamma\beta\alpha}-A^{\alpha\beta\gamma},$
(55) $\displaystyle
0=M^{\alpha\beta\gamma}_{k\,1}=-kA^{\alpha\beta\gamma}+A^{\beta\alpha\gamma}+A^{\gamma\beta\alpha}+A^{\beta\gamma\alpha},$
$\displaystyle
0=M^{\alpha\beta\gamma}_{1\,k}=-A^{\alpha\beta\gamma}+kA^{\beta\alpha\gamma}-A^{\gamma\alpha\beta}-A^{\alpha\gamma\beta},$
$\displaystyle
0=M^{\alpha\beta\gamma}_{k\,0}=-kD\big{[}A^{\alpha\beta\gamma}\big{]}+D\big{[}A^{\gamma\beta\alpha}\big{]}+T^{\beta\gamma\alpha}-T^{\alpha\beta\gamma}.$

The first equation from (55) yields
$A^{\gamma\beta\alpha}=A^{\alpha\beta\gamma}$. Note, this holds for all
$\alpha,\beta,\gamma$. The second and third equations together give

$\displaystyle 0$
$\displaystyle=M^{\alpha\beta\gamma}_{k\,1}+M^{\alpha\beta\gamma}_{1\,k}=$
$\displaystyle=-kA^{\alpha\beta\gamma}+A^{\beta\alpha\gamma}+A^{\gamma\beta\alpha}+A^{\beta\gamma\alpha}-A^{\alpha\beta\gamma}+kA^{\beta\alpha\gamma}-A^{\gamma\alpha\beta}-A^{\alpha\gamma\beta}=$
$\displaystyle=(k+1)\big{(}A^{\beta\alpha\gamma}-A^{\alpha\beta\gamma}\big{)}$

Here we used $A^{\beta\gamma\alpha}=A^{\alpha\gamma\beta}$, which follows from
the eailier as $\alpha,\beta,\gamma$ are arbitrary.

So, we have shown, that $A^{\alpha\beta\gamma}=A^{\beta\alpha\gamma}$ and
$A^{\alpha\beta\gamma}=A^{\gamma\beta\alpha}$. The permutation group in three
elements is generated by arbitrary two transpositions, thus, the corresponding
object is totally symmetric in upper indices.

Finally, the last equation of (55) yields
$(1-k)D\big{(}A^{\alpha\beta\gamma}\big{)}=0$. As $A^{\alpha\beta\gamma}$ is a
differential polynomial of degree zero, which does not depend on $x$, we get
that it is constant. ∎

Due to the symmetry of $A^{\alpha\beta\gamma}$, the second equation in (55)
takes form

$0=(3-k)A^{\alpha\beta\gamma}.$

Thus, for $k\geq 5$ we get that $A^{\alpha\beta\gamma}$ identically vanish.
Recall, that
$A^{\alpha\beta\gamma}=\Gamma^{\alpha\beta}_{s}h^{s\gamma}=g^{\alpha
q}\Gamma^{\beta}_{qs}h^{s\gamma}$. As both $h$ and $g$ are non-degenerate, we
get, that $\Gamma^{\beta}_{qs}$ identically vanish and in coordinates
$u^{1},\dots,u^{n}$ both metrics are in constant form.

We have shown, that the statement of the Lemma 4.1 follows from conditions
(54), which, in turn, are equivalent to the compatibility of $\mathcal{A}_{g}$
and $\mathcal{B}_{h}$. Let us show, that in case $k=3$ the statement of the
Lemma 4.1 is, in fact, equivalent to the compatibility. The compatibility
condition in the form (52) is written as

$\displaystyle 0=$ $\displaystyle
A^{\alpha\beta\gamma}\Bigg{(}\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}^{(3)}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}^{(3)}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-$
$\displaystyle-$
$\displaystyle\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\gamma}\mathcal{b}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}\mathcal{a}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}^{(1)}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x\Bigg{)}=$
$\displaystyle=$ $\displaystyle
A^{\alpha\beta\gamma}\Bigg{(}-\int\limits^{2\pi}_{0}\mathcal{a}^{(4)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-3\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-3\int\limits^{2\pi}_{0}\mathcal{a}^{(2)}_{\alpha}\mathcal{b}^{(2)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+$
$\displaystyle+\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+3\int\limits^{2\pi}_{0}\mathcal{a}^{(2)}_{\alpha}\mathcal{b}^{(2)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+3\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}\mathcal{a}_{\alpha}\mathcal{b}^{(4)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+$
$\displaystyle+\int\limits^{2\pi}_{0}\mathcal{a}^{(4)}_{\alpha}\mathcal{b}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}_{\alpha}\mathcal{b}^{(4)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x+$
$\displaystyle+\int\limits^{2\pi}_{0}\mathcal{a}^{(3)}_{\alpha}\mathcal{b}^{(1)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x-\int\limits^{2\pi}_{0}\mathcal{a}^{(1)}_{\alpha}\mathcal{b}^{(3)}_{\beta}\mathcal{c}_{\gamma}\mathrm{d}x\Bigg{)}=0.$

Again, we recall, that
$A^{\alpha\beta\gamma}=\Gamma^{\alpha\beta}_{s}h^{s\gamma}$ and $h^{s\gamma}$
is in constant form. Thus, $\Gamma^{\alpha\beta}_{s}$ are constant. As
$A^{\alpha\beta\gamma}=A^{\beta\alpha\gamma}$ we get, that
$\Gamma^{\alpha\beta}_{s}$ are also symmetric in upper indices.

We denote $a^{\alpha\beta}_{s}=-\Gamma^{\alpha\beta}_{s}$. The condition (48)
on contrvariant Christoffel symbols implies that
$g^{\alpha\beta}=2(b^{\alpha\beta}+a^{\alpha\beta}_{s}u^{s})$. Here we denoted
the constant term as $2b^{\alpha\beta}$, this is exactly the matrix of
$g^{\alpha\beta}$ at the coordinate origin, thus, it is non-degenerate. The
second condition (49) yields relation
$a^{\alpha\beta}_{q}(b^{q\gamma}+a^{q\gamma}_{s}u^{s})=a^{\gamma\beta}_{q}(b^{q\alpha}+a^{q\alpha}_{s}u^{s})$.
This yields two conditions

$a^{\alpha\beta}_{q}b^{q\gamma}=a^{\gamma\beta}_{q}b^{q\alpha},\quad
a^{\alpha\beta}_{q}a^{q\gamma}_{s}=a^{\gamma\beta}_{q}a^{q\alpha}_{s}.$ (56)

Finally, the symmetry $A^{\alpha\beta\gamma}=A^{\gamma\beta\alpha}$ yields
$a^{\alpha\beta}_{q}h^{q\gamma}=a^{\gamma\beta}_{q}h^{q\alpha}$. Thus, we have
shown that the conditions of Theorem 2.2 are necessary.

For $k\geq 5$ it is obvious, that these conditions are also sufficient. For
$k=3$ the sufficiency can be found in the literature (see [21],[22]), but to
keep our work self-contained we prove this fact.

We say that coordinates $u^{1},\dots,u^{n}$ are Frobenius coordinates for
contrvariant metric $g^{\alpha\beta}$ if

$g^{\alpha\beta}=2(b^{\alpha\beta}+a^{\alpha\beta}_{s}u^{s}),$ (57)

where constants $b^{\alpha\beta},a^{\alpha\beta}_{s}$ satisfy (56). Note that
by construction, the constants are both symmetric in upper indices. Recall
that the name comes from the fact that $a^{\alpha\beta}_{s}$ define
commutative associative algebra with Frobenius form $b^{\alpha\beta}$. The
next Lemma proves a stronger result, then the one we need.

######  Lemma 4.2.

Let $g$ be a contrvariant metric and $u^{1},\dots,u^{n}$ a coordinate system.
The following two conditions are equivalent:

  1. 1.

In coordinates $u^{1},\dots,u^{n}$ the contrvariant Christoffel symbols of $g$
are constant and symmetric in upper indices

  2. 2.

The metric $g^{\alpha\beta}$ is in Frobenius coordinates.

If either of these conditions hold, the metric is flat and
$-\Gamma^{\alpha\beta}_{s}=a^{\alpha\beta}_{s}$.

###### Proof.

We have already proven that in the statement of the Lemma condition 1 implies
condition 2. Now, let us show the inverse.

In given coordinates define Christoffel symbols of the Levi-Civita connection
of $g^{\alpha\beta}$ as

$\Gamma^{\beta}_{rs}=-g_{rq}a^{q\beta}_{s}.$

By construction
$\nabla_{s}g^{\alpha\beta}=2a^{\alpha\beta}_{s}-a^{\alpha\beta}_{s}-a^{\beta\alpha}_{s}=0$.
At the same time conditions (56) imply, that
$a^{\alpha\beta}_{s}g^{s\gamma}=g^{\alpha
r}\Gamma^{\beta}_{rs}g^{s\gamma}=g^{\alpha r}\Gamma^{\beta}_{sr}g^{s\gamma}$.
Thus, the connection is symmetric and $g$ is parallel along it. Thus, it
implies that the corresponding connection is Levi-Civita connection and
$a^{\alpha\beta}_{s}=-\Gamma^{\alpha\beta}_{s}$. Thus, condition 2 of Lemma
implies condition 1.

Finally, we apply the general formula for Riemann tensor of $g$ in terms of
contrvariant Christoffel symbols (formula 0.9 in [24]):

$R^{\beta\gamma\alpha}_{s}=-\frac{\partial\Gamma^{\gamma\alpha}_{p}}{\partial
u^{s}}g^{p\beta}+\frac{\partial\Gamma^{\gamma\alpha}_{s}}{\partial
u^{p}}g^{p\beta}+\Gamma^{\beta\gamma}_{q}\Gamma^{q\alpha}_{s}-\Gamma^{\beta\alpha}_{q}\Gamma^{q\gamma}_{s}$

The contrvariant Christoffel symbols are constants. We get
$R^{\beta\gamma\alpha}_{s}=a^{\beta\gamma}_{q}a^{q\alpha}_{s}-a^{\beta\alpha}_{q}a^{q\gamma}_{s}=a^{\gamma\beta}_{q}a^{q\alpha}_{s}-a^{\alpha\beta}_{q}a^{q\gamma}_{s}=0$.
The last condition is the second condition from (56) and here we used symmetry
of $a^{\alpha\beta}_{s}$ in upper indices. The Lemma is proved. ∎

The Lemma 4.2 implies that the conditions of Theorem 2.2 for $k=3$ are
sufficient. Now, let us prove Corollary 2.1.

By construction $S^{\alpha\beta}_{s}=g^{\alpha
q}\big{(}\Gamma^{\beta}_{qs}-\bar{\Gamma}^{\beta}_{qs}\big{)}$. Fix Darboux
coordinates $u^{1},\dots,u^{n}$ of $\mathcal{B}_{h}$. We get that in given
coordinates
$\Gamma^{\alpha\beta}_{s}=S^{\alpha\beta}_{s}=-a^{\alpha\beta}_{s}$. This
implies that the conditions of Corollary 2.1 are equivalent to those of
Theorem 2.2 in Darboux coordinates.

Both Theorem 2.2 and Corollary 2.1 are proved.

##  5 Proof of Theorem 2.3

We start with trivial case: the classification of pairs of bilinear forms over
$\mathbb{R}$ can be found in [19] (Theorem 9.2). Each pair
$h^{\alpha\beta},b^{\alpha\beta}$ of non-degenerate forms in dimension two can
be brought to one of the following form

$\displaystyle\left\\{\left(\begin{array}[]{cc}h_{1}&0\\\
0&h_{2}\end{array}\right),\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\right\\},\left\\{\left(\begin{array}[]{cc}h_{1}&0\\\
0&h_{2}\end{array}\right),\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)\right\\},\left\\{\left(\begin{array}[]{cc}h_{1}&0\\\
0&h_{2}\end{array}\right),\left(\begin{array}[]{cc}-1&0\\\
0&-1\end{array}\right)\right\\},$
$\displaystyle\left\\{\left(\begin{array}[]{cc}0&h_{1}\\\
h_{1}&1\end{array}\right),\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\right\\},\left\\{\left(\begin{array}[]{cc}-h_{2}&h_{1}\\\
h_{1}&h_{2}\end{array}\right),\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\right\\}$

In each family, different collections of constants $h_{1},h_{2}$ provide
pairs, which cannot be transformed one into the other by coordinate change.
This provides families $(01)$ through $(05)$ in the statement of Theorem 2.3.

We will need the following reformulation of (23). In a given basis, consider a
pair of operators $R_{1}\eta=\eta\star\eta_{1}$ and
$R_{2}\eta=\eta\star\eta_{2}$. Bilinear form $h^{\alpha\beta}$ with matrix $H$
is invariant form if and only if both operators are self-adjoint with respect
to this form, that is

$R_{1}^{T}H=HR_{1},\quad R_{2}^{T}H=HR_{2}.$

We proceed as follows: for each algebra from the list (24) we explicitly write
$R_{1},R_{2}$ and $h^{\alpha\beta}$, calculate using the above observation (we
omit the calculations itself).

Then we describe the coordinate changes that preserve the structure relation
of the algebras from (24). Using these coordinate changes, we bring $h$ to
some normal form. In case $\mathfrak{a}_{i},i\geq 2$ the corresponding
coordinate system is unique and, thus, different collection of constants
$b_{j},h_{j},j=1,2$ provide non-equivalent normal forms of inhomogeneous
Hamiltonian operators of type $1+3$.

In case of $\mathfrak{a}_{1}$ there are two canonical coordinate systems, but
the corresponding coordinate changes preserve the form of $b^{\alpha\beta}$,
thus, the normal form is unique.

Case $\mathfrak{a}_{1}$:  In a given basis $\eta^{1},\eta^{2}$ the only non-
zero structure relation of this algebra is

$\eta^{2}\star\eta^{2}=\eta^{2}.$

The operators $R_{1},R_{2}$ and invariant form $h^{\alpha\beta}$ are

$R_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right),\quad
R_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right),\quad
h^{\alpha\beta}=\left(\begin{array}[]{cc}h_{1}&0\\\
0&h_{2}\end{array}\right).$

The coordinate change that preserves the structure relations of
$\mathfrak{a}_{1}$ is

$\bar{\eta}^{1}=\alpha\eta^{1},\quad\bar{\eta}^{2}=\eta^{2}.$

Under this transformation the coefficient $h_{1}$ can be made into either $1$
or $-1$. The corresponding coordinate system is not uniquely defined. Besides
trivial coordinate change we have one more:
$\bar{\eta}^{1}=-\eta^{1},\bar{\eta}^{2}=\eta^{2}$. At the same time this
coordinate change does not change the normal form, so we get two normal forms
$(11)$ and $(12)$.

Case $\mathfrak{a}_{2}$: In a given basis $\eta^{1},\eta^{2}$ the only non-
zero structure relation of this algebra is

$\eta^{2}\star\eta^{2}=\eta^{1}.$

The operators $R_{1},R_{2}$ and invariant form $h^{\alpha\beta}$ are

$R_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right),\quad
R_{2}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right),\quad
h^{\alpha\beta}=\left(\begin{array}[]{cc}0&h_{1}\\\
h_{1}&h_{2}\end{array}\right).$

The coordinate change that preserves the structure relation of
$\mathfrak{a}_{2}$ is

$\bar{\eta}^{1}=\alpha^{2}\eta^{1},\quad\bar{\eta}^{2}=\beta\eta^{1}+\alpha\eta^{2}.$
(58)

Taking coordinate change in the form

$\bar{\eta}^{1}=\frac{1}{(h_{1})^{2/3}}\eta^{1},\quad\bar{\eta}^{2}=-\frac{h_{2}}{2(h_{1})^{4/3}}\eta^{1}+\frac{1}{(h_{1})^{1/3}}\eta^{2}$

we bring $h^{\alpha\beta}$ to the form $h_{1}=1,h_{2}=0$.

Coordinate change 58 that preserves $h^{\alpha\beta}$, satisfies conditions
$h(\bar{\eta}^{1},\bar{\eta}^{2})=\alpha^{3}=1$ and
$h(\bar{\eta}^{2},\bar{\eta}^{2})=2\alpha\beta=0$. This implies that the only
such coordinate change is trivial one. Thus, the canonical coordinate system
is unique and we obtain the normal form $(21)$.

Case $\mathfrak{a}_{3}$: In a given basis $\eta^{1},\eta^{2}$ the non-zero
structure relations of this algebra are

$\eta^{2}\star\eta^{2}=\eta^{2},\quad\eta^{1}\star\eta^{2}=\eta^{1}.$

The operators $R_{1},R_{2}$ and invariant form $h^{\alpha\beta}$ are

$R_{1}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right),\quad
R_{2}=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\quad
h^{\alpha\beta}=\left(\begin{array}[]{cc}0&h_{1}\\\
h_{1}&h_{2}\end{array}\right).$

The coordinate change that preserves the structure relations of
$\mathfrak{a}_{3}$ is

$\bar{\eta}^{1}=\alpha\eta^{1},\quad\bar{\eta}^{2}=\eta^{2}.$

We take $\alpha=\frac{1}{h_{1}}$ and get the invariant form with $h_{1}=1$.
Following the same argument as in previous case, we get that the corresponding
coordinate system is unique. We get the normal form $(31)$.

Case $\mathfrak{a}_{4}$: In a given basis $\eta^{1},\eta^{2}$ the non-zero
structure relations of this algebra are

$\eta^{1}\star\eta^{1}=\eta^{1},\quad\eta^{2}\star\eta^{2}=\eta^{2}.$

In given coordinates operators $R_{1},R_{2}$ and invariant form
$h^{\alpha\beta}$ are

$R_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right),\quad
R_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right),\quad
h^{\alpha\beta}=\left(\begin{array}[]{cc}h_{1}&0\\\
0&h_{2}\end{array}\right).$

The only coordinate change that preserves the structure relations of algebra
$\mathfrak{a}_{4}$ is the trivial one. Thus, the corresponding coordinate
system is unique and we obtain the normal form $(41)$.

Case $\mathfrak{a}_{5}$: In a given basis $\eta^{1},\eta^{2}$ the non-zero
structure relations of this algebra are

$\quad\quad\eta^{2}\star\eta^{2}=\eta^{2},\quad\eta^{1}\star\eta^{1}=-\eta^{2},\quad\eta^{1}\star\eta^{2}=\eta^{1}..$

The operators $R_{1},R_{2}$ and invariant form $h^{\alpha\beta}$ are

$R_{1}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right),\quad
R_{2}=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\quad
h^{\alpha\beta}=\left(\begin{array}[]{cc}-h_{2}&h_{1}\\\
h_{1}&h_{2}\end{array}\right).$

The only coordinate change that preserves the structure relations of algebra
$\mathfrak{a}_{5}$ is the trivial one. Same as in the previous case, the
corresponding coordinate system is unique. We get the normal form $(51)$. The
Theorem is proved.

##  6 Proof of Theorem 2.4

We start with several algebraic lemmas.

######  Lemma 6.1.

Consider vector space $V$ of dimension $n$ and its dual $V^{*}$. Fix a basis
$\eta^{1},\dots,\eta^{n}$ in $V^{*}$ and the dual basis in $V$. Assume that in
given basis the collection of constants $a^{\alpha\beta}_{s}$ defines the
structure of a commutative associative algebra. Assume also that
$b^{\alpha\beta}$ is Frobenius form of this algebra. Then

$c^{\beta}_{rs}=\frac{1}{2}b_{rq}a^{q\beta}_{s}$

defines the structure of a commutative associative algebra on $V$ in the dual
basis. In other words

$c^{\beta}_{ps}=c^{\beta}_{sp},\quad
c^{\beta}_{sq}c^{q}_{pr}=c^{q}_{sp}c^{\beta}_{qr}.$

This algebra is isomorphic to the one on $V^{*}$. In particular, if $e\in
V^{*}$ is unity of $a^{\alpha\beta}_{s}$ with coordinates $e_{1},\dots,e_{n}$,
then $f\in V$ with coordinates $f^{p}=2b^{pq}e_{q}$ is unity of
$c^{\beta}_{ps}$.

###### Proof.

Symmetric non-degenerate bilinear form $2b^{\alpha\beta}$ can be understood as
invertible map $b:V^{*}\to V$. We get

$\frac{1}{2}b_{sp}b_{rq}a^{pq}_{m}b^{m\beta}=\frac{1}{2}b_{sp}b_{rq}a^{\beta
q}_{m}b^{mp}=\frac{1}{2}b_{rq}a^{\beta
q}_{m}\delta^{m}_{s}=\frac{1}{2}b_{rq}a^{q\beta}_{s}=c^{\beta}_{rs}.$

This implies that $c^{\beta}_{rs}$ is commutative associative algebra,
isomorphic to the one defined by $a^{\alpha\beta}_{s}$ on $V^{*}$. It also
implies the statement about the unity. The Lemma is proved. ∎

The next fact from linear algebra is well-known, but to keep our work self-
contained we provide it with proof.

######  Lemma 6.2.

Consider a real vector space $V$ of dimension $n$ and a non-degenerate linear
operator $R:V\to V$. Then there exists a polynomial $p(t)$ (possibly with
complex coefficients), such that for $L=p(R)$ one has $L^{2}=R$.

###### Proof.

First, assume that $R$ has no Jordan blocks, then one can take $p(t)$ to be
the Lagrange polynomial for square roots of eigenvalues. Here the complex
coefficients may appear, for example, for negative real eigenvalues.

Now assume that $R$ is in general form. Consider its Jordan-Chevalley
decomposition $R=R_{s}+R_{n}$, where $R_{s},R_{n}$ are semisimple and
nilpotent parts respectively. Moreover, there exist polynomials $q_{s}(t)$ and
$q_{n}(t)$ both without constant terms, such that
$R_{s}=q_{s}(R),R_{n}=q_{n}(R)$ (Proposition p.17, [25]). We rewrite it as
$R=R_{s}(\operatorname{Id}+R^{-1}_{s}R_{n})$. We get

$N=R^{-1}_{s}R_{n}=\Big{(}q_{s}(R)\Big{)}^{-1}q_{n}(R).$

The inverse of a matrix $R^{-1}_{s}$ can be written as a polynomial, thus, we
get that $N$ is a polynomial in $R$.

By earlier argument there exists a good root of $R_{s}$, which is expressed as
a polynomial of $R_{s}$ and, thus, as a polynomial of $R$. We denote it as
$R^{1/2}_{s}$.

As $N$ is nilpotent $(\operatorname{Id}+N)^{-1/2}$ is a polynomial in $R$.
Finally we take

$L=R^{1/2}_{s}(\operatorname{Id}+N)^{-1/2}.$

to be a good root of $R$ in general form. ∎

######  Lemma 6.3.

Assume that $c^{\beta}_{ps}$ and $r^{\beta}_{p}$ are defined as in the
statement of Theorem 2.4. Consider $l^{\beta}_{q}$ to be the components of
good square root of $r^{\alpha}_{p}$ and $\bar{l}^{\beta}_{q}$ the components
of the inverse of the good square root. The identity

$m^{\beta}_{r}c^{r}_{ps}=c^{\beta}_{pr}m^{r}_{s}=m^{r}_{p}c^{\beta}_{rs}$ (59)

holds for both $m^{\alpha}_{p}=l^{\alpha}_{p}$ and
$m^{\alpha}_{p}=\bar{l}^{\alpha}_{p}$.

###### Proof.

First, let us check that (59) holds for $m^{\alpha}_{p}=r^{\alpha}_{p}$. The
constants $a^{\alpha\beta}_{s},b^{\alpha\beta},h^{\alpha\beta}$ are the ones
defined in the statement of Theorem 2.4. By definition of Frobenius forms we
get

$\displaystyle h^{\alpha s}a_{s}^{\beta\gamma}=h^{\alpha
s}b_{sm}b^{mr}a^{\beta\gamma}_{r}=2r^{\alpha}_{m}b^{\gamma r}a^{\beta m}_{r}$

Lowering index $\gamma$ with $b$ and renaming the rest of the indices we
arrive to the identity

$h^{\alpha s}c^{\beta}_{ps}=2r^{\alpha}_{m}a^{\beta m}_{s}.$

At the same time, as $h^{\alpha\beta}$ is invariant and $a^{\alpha\beta}_{s}$
is commutative, we get

$2r^{\alpha}_{m}a^{\beta m}_{s}=h^{\alpha s}c^{\beta}_{ps}=h^{\beta
s}c^{\alpha}_{ps}.$

Lowering $\beta$ with $b$, and, again, renaming the indices we get

$2r^{\alpha}_{m}c^{m}_{ps}=c^{\alpha}_{pm}r^{m}_{s}$

We know that $c^{\beta}_{ps}=c^{\beta}_{sp}$ (Lemma 6.1), so $r^{\alpha}_{p}$
satisfies (59).

Now recall that if identity (59) holds for $r^{\alpha}_{p}$, then it holds for
all powers of $r^{\alpha}_{p}$ and, thus, for all polynomials of the operator.
As $l$ is a good root, then the identity (59) holds for $l^{\alpha}_{p}$. The
inverse of a non-degenerate matrix is a polynomial of this matrix, thus, by
the same argument the identity (59) holds for $\bar{l}^{\alpha}_{p}$. ∎

######  Lemma 6.4.

Consider a differential polynomial $\mathcal{h}$ and assume that
$D^{j}(\mathcal{h})=0$ for some $j\geq 1$. Then $\mathcal{h}$ is a constant.

###### Proof.

It is enough to prove this for a homogeneous differential polynomial of degree
$m$. Let us order the terms with respect to ”algebraic degree”, that is the
number of symbols $u^{\alpha}_{x^{j}}$ for $j\geq 1$.

We denote the decomposition of $\mathcal{h}$ into the sum of homogeneous (with
respect to this ordering) terms as $\mathcal{h}=\mathcal{h}_{(i)}+\dots$,
where $1\leq i\leq m$ and $\mathcal{h}_{(i)}\neq 0$. By construction
$D(\mathcal{h}_{(i)})=\bar{\mathcal{h}}_{(i)}+\bar{\mathcal{h}}_{(i+1)}$ and
functional coefficients of $\bar{\mathcal{h}}_{(i)}$ coincide with functional
coefficients of $\mathcal{h}_{(i)}$.

Applying $j$ times $D$ we get that vanishing of the result implies vanishing
of the term of order $(i)$. This, in turn, means, that $\mathcal{h}_{(i)}=0$.
This contradiction completes the proof. ∎

Now let us proceed to the proof of the Theorem.

First, let us discuss the solvability of (26). The constants $c^{\beta}_{ps}$
are symmetric in lower indices (Lemma 6.1). From (26), the formulas for the
components $\mathcal{v}_{i}^{\alpha}$ of series $\mathrm{v}^{\alpha}$ are

$\displaystyle u^{\beta}$
$\displaystyle=\mathcal{v}_{1}^{\beta}+\frac{1}{2}c^{\beta}_{ps}\mathcal{v}_{1}^{p}\mathcal{v}_{1}^{s},$
(60)
$\displaystyle\Big{(}\delta^{\beta}_{q}+c^{\beta}_{qs}\mathcal{v}^{s}_{1}\Big{)}\mathcal{v}^{q}_{i}$
$\displaystyle=-\frac{1}{2}\sum\limits_{j=2}^{i-1}c^{\beta}_{ps}\mathcal{v}_{j}^{p}\mathcal{v}_{i-j}^{s}+l^{\beta}_{s}(\mathcal{v}^{s}_{i-1})_{x},\quad
i\geq 2$

The first formula is actually Balinskii-Novikov formula from [21]. It is
written in the form
$u^{\alpha}=F^{\alpha}(\mathcal{v}^{1},\dots,\mathcal{v}^{n})$. At the point
$u^{\alpha}=0,\mathcal{v}^{\beta}_{1}=0$ we have $\frac{\partial
F^{\alpha}}{\partial\mathcal{v}^{q}}=\delta^{\alpha}_{q}$, thus, by the
implicit function theorem in a neighbourhood of this point the solutions
$\mathcal{v}^{\alpha}_{1}(u)$ exist.

For $i\geq 2$ for sufficiently small values of $\mathcal{v}_{1}^{\alpha}$ the
operator $\delta^{\beta}_{q}+\frac{1}{2}c^{\beta}_{qs}\mathcal{v}^{s}_{1}$ on
the l.h.s. of (60) is invertible. Thus, the solution exists and each series is
uniquely defined modulo the choice of solution of Balinskii-Novikov formula.

Fix two discs $\mathrm{M}^{n}$ and $\mathrm{N}^{n}$ with coordinates
$u^{1},\dots,u^{n}$ and $\mathrm{v}^{1},\dots,\mathrm{v}^{n}$. We denote the
r.h.s. of formula 26 as
$\phi(\mathrm{v})=(\phi^{1}(\mathrm{v}),\dots,\phi^{n}(\mathrm{v}))$. For
arbitrary element $c\in\Omega\mathrm{N}^{n}$ we have $\phi\circ
c:\mathbb{R}\to\mathrm{M}^{n}$. That is $\phi$ defines a map
$\phi:\Omega\mathrm{N}^{n}\to\Omega\mathrm{M}^{n}$. This induces the dual map
$\phi^{*}$ on the functionals: for a given density
$\mathcal{h}(u,u_{x},\dots)$ we have

$\phi^{*}\mathcal{H}=\int\limits^{2\pi}_{0}\mathcal{h}(\phi(\mathrm{v}),\phi_{x}(\mathrm{v}),\dots)\mathrm{d}x.$

The density $\mathcal{h}(\phi(\mathrm{v}))$ is not a differential polynomial
in general. It has natural description: it is polynomial in
$\mathrm{v}^{\alpha}_{x^{j}},j=2,\dots$ with coefficients being a smooth
functions of $2n$ variables $\mathrm{v}^{\alpha},\mathrm{v}^{\alpha}_{x}$. So
in this section the polynomiality is not assumed.

The straightforward calculation of variational derivative
$\frac{\delta(\phi^{*}\mathcal{H})}{\delta\mathrm{v}^{s}}$ yields: for
arbitrary $r\in\Omega\mathrm{N}^{n}$ we have

$\displaystyle\frac{\partial}{\partial\epsilon}\int\limits^{2\pi}_{0}\mathcal{h}\big{(}u(\mathrm{v}+\epsilon\mathrm{r},\mathrm{v}_{x}+\epsilon\mathrm{r}_{x})\big{)}\mathrm{d}x=\int\limits^{2\pi}_{0}\sum\limits_{j=0}\Bigg{(}\frac{\partial\mathcal{h}}{\partial
u^{q}_{x^{j}}}D^{j}\Big{(}\frac{\partial
u^{q}}{\partial\mathrm{v}^{s}}r^{s}\Big{)}+\frac{\partial\mathcal{h}}{\partial
u^{q}_{x^{j}}}D^{j}\Big{(}\frac{\partial
u^{q}}{\partial\mathrm{v}_{x}^{s}}r_{x}^{s}\Big{)}\Bigg{)}\mathrm{d}x+\dots=$
$\displaystyle=$
$\displaystyle\int\limits^{2\pi}_{0}\sum\limits_{j=0}\Bigg{(}\frac{\partial\mathcal{h}}{\partial
u^{q}_{x^{j}}}D^{j}\Big{(}\big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\big{)}r^{s}\Big{)}+\frac{\partial\mathcal{h}}{\partial
u^{q}_{x^{j}}}D^{j+1}(l^{q}_{s}r^{s})\Bigg{)}\mathrm{d}x+\dots=$
$\displaystyle=$
$\displaystyle\int\limits^{2\pi}_{0}\Bigg{(}\frac{\delta\mathcal{h}}{\delta
u^{q}}\big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\big{)}r^{s}+l^{q}_{s}\frac{\delta\mathcal{h}}{\delta
u^{q}}r^{s}_{x}\Bigg{)}\mathrm{d}x+\dots=\int\limits^{2\pi}_{0}\Bigg{(}\frac{\delta\mathcal{h}}{\delta
u^{q}}\big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\big{)}-l^{q}_{s}D\Big{(}\frac{\delta\mathcal{h}}{\delta
u^{q}}\Big{)}\Bigg{)}r^{s}\mathrm{d}x+\dots$

The calculation obviously holds for the densities
$\mathrm{v}=\mathcal{v}_{1}+\mathcal{v}_{2}+\dots$, where $\mathcal{v}_{i}$ is
either zero ore differential polynomial of degree $i-1$ in
$u^{\alpha}_{x^{j}},j\geq 0$. For functional
$\mathrm{V}^{\alpha}=\int^{2\pi}_{0}\mathrm{v}^{\alpha}\mathrm{d}x$, where
$\mathrm{v}^{\alpha}$ is one of the solutions of (26), we get identity

$\frac{\delta\mathrm{V}^{\alpha}}{\delta\mathrm{v}^{s}}=\delta^{\alpha}_{s}=\Big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}-l^{q}_{s}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}.$ (61)

Recall, that identity (26) provides a recursion formula for calculation of the
components $\mathcal{v}_{i}$ of the series
$\mathrm{v}=\mathcal{v}_{1}+\mathcal{v}_{2}+\mathcal{v}_{3}+\dots$. The
identity (61), in turn, provides the recursion formulas for calculation of the
components $\frac{\delta\mathcal{v}^{\alpha}_{i}}{\delta u^{s}}$ of the series
$\frac{\delta\mathrm{v}^{\alpha}}{\delta
u^{s}}=\frac{\delta\mathcal{v}^{\alpha}_{1}}{\delta
u^{s}}+\frac{\delta\mathcal{v}^{\alpha}_{2}}{\delta u^{s}}+\dots$.

Now we are going to show that together these two identities imply the first
statement of the theorem by a straightforward computation. Recall that
$\bar{l}^{\beta}_{s}$ stands for the inverse of the operator $l^{\beta}_{s}$.
We rewrite (26) and (61) as

$\displaystyle D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{s}}\Big{)}=\Big{(}\bar{l}^{m}_{s}+\bar{l}^{r}_{p}c^{m}_{pr}\mathrm{v}^{p}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}-\bar{l}^{\alpha}_{s},$ (62)
$\displaystyle\mathrm{v}^{\alpha}_{x}=\bar{l}^{\alpha}_{m}u^{m}-\bar{l}^{\alpha}_{m}\mathrm{v}^{m}-\frac{1}{2}\bar{l}^{\alpha}_{m}c^{m}_{ps}\mathrm{v}^{p}\mathrm{v}^{s}.$

Applying $D$ to both sides of (61) we get

$0=c^{q}_{ps}\mathrm{v}^{p}_{x}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+\Big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\Big{)}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}-l^{m}_{s}D^{2}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}.$

Substituting (62) into previous equation yields

$0=c^{e}_{sq}\bar{l}^{q}_{m}\Big{(}u^{m}-\mathrm{v}^{m}-\frac{1}{2}c^{m}_{pr}\mathrm{v}^{p}\mathrm{v}^{r}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{e}}+\Big{(}\delta^{q}_{s}+c^{q}_{ps}\mathrm{v}^{p}\Big{)}\bar{l}^{m}_{q}\Big{(}\delta^{e}_{m}+c^{e}_{rm}\mathrm{v}^{r}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{e}}-\bar{l}^{\alpha}_{s}-\bar{l}^{\alpha}_{m}c^{m}_{ps}\mathrm{v}^{p}-l^{m}_{s}D^{2}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}.$

We multiply both sides by $l^{\alpha}_{q}$ and apply Lemma 6.3. This yields

$0=c^{q}_{sp}u^{p}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+\Big{(}\delta^{m}_{s}+c^{m}_{ps}\mathrm{v}^{p}+c^{q}_{ps}c^{m}_{qr}\mathrm{v}^{p}\mathrm{v}^{r}-\frac{1}{2}c^{m}_{sq}c^{q}_{pr}\mathrm{v}^{p}\mathrm{v}^{r}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}-\delta^{\alpha}_{s}-c^{\alpha}_{ps}\mathrm{v}^{p}-r^{m}_{s}D^{2}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}.$

Using identity $c^{q}_{ps}c_{qr}^{m}=c_{sq}^{m}c^{q}_{pr}$ from Lemma 6.1, we
finally get

$0=c^{q}_{sp}u^{p}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{s}}+\Big{(}c^{m}_{ps}\mathrm{v}^{p}+\frac{1}{2}c^{m}_{sq}c^{q}_{pr}\mathrm{v}^{p}\mathrm{v}^{r}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}-\delta^{\alpha}_{s}-c^{\alpha}_{ps}\mathrm{v}^{p}-r^{m}_{s}D^{2}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}$

Now we again apply $D$ to both sides. This yields

$\displaystyle 0$
$\displaystyle=c^{q}_{sp}u^{p}_{x}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+c^{q}_{sp}u^{p}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}+D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{s}}\Big{)}+\Big{(}c^{m}_{ps}\mathrm{v}_{x}^{p}+c^{m}_{sq}c^{q}_{pr}\mathrm{v}_{x}^{p}\mathrm{v}^{r}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}+$
$\displaystyle+\Big{(}c^{m}_{ps}\mathrm{v}^{p}+\frac{1}{2}c^{m}_{sq}c^{q}_{pr}\mathrm{v}^{p}\mathrm{v}^{r}\Big{)}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}-c^{\alpha}_{ps}\mathrm{v}^{p}_{x}-r^{m}_{s}D^{3}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}$

Here we have again used identities for $c^{\beta}_{ps}$ from Lemma 6.1.
Regrouping terms we get

$\displaystyle 0$
$\displaystyle=c^{q}_{sp}u^{p}_{x}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+c^{q}_{sp}u^{p}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}+D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{s}}\Big{)}+c^{r}_{sp}\mathrm{v}^{p}_{x}\Bigg{(}\Big{(}\delta^{m}_{r}+c^{m}_{rq}\mathrm{v}^{q}\Big{)}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}-\delta^{\alpha}_{r}\Bigg{)}+$
$\displaystyle+\Big{(}c^{m}_{ps}\mathrm{v}^{p}+\frac{1}{2}c^{m}_{sq}c^{q}_{pr}\mathrm{v}^{p}\mathrm{v}^{r}\Big{)}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}-r^{m}_{s}D^{3}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}$

From the first formula of (62) we get that the expression in big brackets
equals to $l_{s}^{m}D\big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\big{)}$. Regrouping coefficients and applying (26) we get

$0=c^{q}_{sp}u^{p}_{x}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}+2c^{q}_{sp}u^{p}D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}+D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{s}}\Big{)}-r^{m}_{s}D^{3}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{m}}\Big{)}.$

Raising the indices by $2b^{\alpha q}$ we get

$2(b^{\alpha q}+a^{\alpha
q}_{s}u^{s})D\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}+a^{\alpha\beta}_{s}u^{s}_{x}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}=h^{\alpha q}D^{3}\Big{(}\frac{\delta\mathrm{V}^{\alpha}}{\delta
u^{q}}\Big{)}$ (63)

Now recall that if $\mathcal{v}_{i}$ is a differential polynomial of degree
$i-1$, then its partial variational derivative
$\frac{\delta\mathcal{v}_{i}}{\delta u^{\alpha}}$ is also a differential
polynomial of degree $i-1$. At the same time operator $D$ increases the degree
of differential polynomial by one. This observation implies that after
substituting
$\mathrm{v}^{\alpha}=\mathcal{v}^{\alpha}_{1}+\mathcal{v}^{\alpha}_{2}+\dots$
into (63) we get two Harry Dym hierarchies

$\begin{array}[]{l}\mathcal{A}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{1}}{\delta u^{q}}=0,\\\ \mathcal{A}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{2i+1}}{\delta u^{q}}=\mathcal{B}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{2i+1}}{\delta u^{q}}\end{array}\quad\text{
and }\quad\begin{array}[]{l}\mathcal{A}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{2}}{\delta u^{q}}=0,\\\ \mathcal{A}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{2i+2}}{\delta u^{q}}=\mathcal{B}^{\beta
q}\frac{\delta\mathcal{V}^{\alpha}_{2i+1}}{\delta
u^{q}}\end{array}\quad\text{i = 1, \ldots}.$

Here $\mathcal{V}^{\alpha}_{i}=\int^{2\pi}_{0}\mathcal{v}_{i}\mathrm{d}x$.
After renaming $\mathcal{h}_{i}=\mathcal{v}_{2i-1}$ for $i=1,\dots$ the first
hierarchy produces the first statement of Theorem 2.4. We will need the
following Lemma.

As for the second hierarchy taking flat coordinates for $\mathcal{A}_{g}$, we
get that by Lemma 6.4 the $\frac{\delta\mathcal{V}_{i}}{\delta u^{q}}=0$ for
all even $i$. Thus, the second hierarchy is trivial and $\mathcal{v}_{i}$ for
even $i$ are total derivatives.

Recall that $\mathcal{A}_{g}$ and $\mathcal{B}_{h}$ are compatible Poisson
brackets. The constructed solutions provide the Lenard-Magri chains for
corresponding bracket. Thus, the second statement of Theorem 2.4 follows from
general bihamiltonian formalism for such brackets.

Now assume that we start Harry Dym hierarchy with
$\mathcal{V}^{\alpha}_{1}=0$. The second equation of the hierarchy becomes

$\mathcal{B}^{\beta q}D^{3}\Big{(}\frac{\delta\mathcal{H}^{\alpha}_{2}}{\delta
u^{q}}\Big{)}=0.$

By Lemma 6.4 this implies that in the case of homogeneous differential
polynomial densities, the only solution is $\mathcal{H}^{\alpha}_{2}=0$.
Applying the same argument, we get that the entire hierarchy is zero. Thus,
the third statement if proved.

To prove the last statement of the theorem we first recall that by Lemma 6.1,
the algebra $c^{\beta}_{ps}$ has unity $f=(f^{1},\dots,f^{n})$. Adding
$\frac{1}{2}f^{\alpha}$ to both sides of (26) we get

$u^{\alpha}+\frac{1}{2}f^{\alpha}=\frac{1}{2}f^{\alpha}+\mathrm{v}^{\alpha}+\frac{1}{2}c^{\beta}_{ps}\mathrm{v}^{p}\mathrm{v}^{s}+l^{\beta}_{s}\mathrm{v}^{s}_{x}=\frac{1}{2}c^{\beta}_{ps}(\mathrm{v}^{p}+f^{p})(\mathrm{v}^{s}+f^{s})+l^{\beta}_{s}(\mathrm{v}^{s}+f^{s})_{x}.$

At the same time we get the following identity

$\frac{1}{2}a^{\beta\alpha}_{q}f^{q}=b^{\alpha
m}c^{\beta}_{mq}f^{q}=b^{\alpha\beta}.$

Thus, the formula for $\mathcal{A}^{\alpha q}_{g}$ takes form

$\mathcal{A}^{\beta q}_{g}=2(b^{\beta q}+a^{\beta q}_{s}u^{s})D+a^{\beta
q}_{s}u^{s}_{x}=2a^{\beta q}_{s}(u^{s}+\frac{1}{2}f^{s})D+a^{\beta
q}_{s}(u^{s}+\frac{1}{2}f^{s})_{x}.$

Shifting $u^{\beta}\to u^{\beta}+\frac{1}{2}f^{\beta}$ we get the fourth
statement of the Theorem.

##  7 Appendix: some facts about commutative associative algebras with
Frobenius forms

In this section we collect some basic facts about commutative associative
algebras and their invariant forms. We always assume that algebra is over
$\mathbb{R}$ or $\mathbb{C}$. If the field is not specified, the result holds
for both.

######  Proposition 7.1.

Consider a commutative associative algebra $\mathfrak{a}$ and arbitrary
$m\in\mathfrak{a}^{*}$. Then $h_{m}(\xi,\eta)=m(\xi\star\eta)$ is an invariant
form. We call this form exact.

###### Proof.

We have

$h_{m}(\xi\star\eta,\zeta)=m((\xi\star\eta)\star\zeta)=m(\xi\star(\eta\star\zeta))=h_{m}(\xi,\eta\star\zeta).$

The Proposition is proved. ∎

Among others we deal with commutative associative algebras with unity

######  Proposition 7.2.

Consider a commutative associative algebra $\mathfrak{a}$. Then

  1. 1.

If $\mathfrak{a}$ has unity, then every invariant form is exact

  2. 2.

If there exists a non-degenerate exact invariant form, then $\mathfrak{a}$ has
unity

###### Proof.

First, assume that $h$ is an invariant form for commutative associative
algebra $\mathfrak{a}$ with unity $e$. Define $m\in\mathfrak{a}^{*}$ as
$m(\eta)=h(e,\eta)$. We have

$h(\xi,\eta)=h(e,\xi\star\eta)=h_{m}(\xi,\eta).$

The first part of Proposition is proved. Now proceed to the second part.
Assume that $h_{m}$ is non-degenerate exact invariant form for
$m\in\mathfrak{a}^{*}$. It can be treated as map
$h_{m}:\mathfrak{a}\to\mathfrak{a}^{*}$. We write

$h_{m}(\xi,\zeta)=m(\xi\star\zeta)=h_{m}(\xi\star\zeta,h_{m}^{-1}m)=h_{m}(\xi\star
h_{m}^{-1}m,\zeta).$ (64)

Denote $\eta=h_{m}^{-1}m$ and consider the operator
$R_{\eta}\xi=\xi\star\eta$. The formula (64) takes form

$h_{m}(\xi,\eta)=h_{m}(R_{\eta}\xi,\zeta)$

By construction $R_{\eta}=h_{m}^{-1}h_{m}=\operatorname{Id}$ and $\eta$ is
unity. The Proposition is proved. ∎

The commutative associative algebra $\mathfrak{a}$ is called reducible if it
can be written as
$\mathfrak{a}=\mathfrak{a}^{\prime}\oplus\mathfrak{a}^{\prime\prime}$ for some
commutative associative algebras
$\mathfrak{a}^{\prime},\mathfrak{a}^{\prime\prime}$, and irreducible
otherwise. By definition, every commutative associative algebra is a sum of
irreducible algebras.

######  Proposition 7.3.

Assume that a commutative associative algebra $\mathfrak{a}$ is reducible,
that is

$\mathfrak{a}=\mathfrak{a}^{\prime}\oplus\mathfrak{a}^{\prime\prime},$ (65)

where $\mathfrak{a}^{\prime},\mathfrak{a}^{\prime\prime}$ are both commutative
associative algebras. If $\mathfrak{a}$ has unity and invariant non-degenerate
form $h$, then both $\mathfrak{a}^{\prime},\mathfrak{a}^{\prime\prime}$ have
unities, decomposition (65) is orthogonal with respect to $h$ and restrictions
of $h$ on each summand are non-degenerate invariant forms.

###### Proof.

Denote unity in $\mathfrak{a}$ as $e$. Consider the decomposition
$e=e^{\prime}+e^{\prime\prime}$ of the unity with respect to (65). By
construction both $e^{\prime},e^{\prime\prime}$ are unities for
$\mathfrak{a}^{\prime},\mathfrak{a}^{\prime\prime}$ respectively.

For arbitrary $\xi\in\mathfrak{a},\eta\in\mathfrak{a}^{\prime\prime}$ we have
$\xi\star\eta=0$ and, thus,

$h(\xi,\eta)=h(\xi,\eta\star e)=h(\xi\star\eta,e)=0.$

Finally, if either of $h^{\prime},h^{\prime\prime}$ has kernel, then this is
the kernel of the entire $h$. The proposition is proved. ∎

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