# Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction

L.V. Bogdanov, Jen-Hsu Chang‡ and Yu-Tung Chen† L.D. Landau ITP, Kosygin str.
2, Moscow 119334, Russia Department of Computer Science, National Defense
University, Taoyuan, Taiwan, E-mail<EMAIL_ADDRESS>${\ddagger}$
corresponding author.

###### Abstract

We study Manakov-Santini equation, starting from Lax-Sato form of associated
hierarchy. The waterbag reduction for Manakov-Santini hierarchy is introduced.
Equations of reduced hierarchy are derived. We construct new coordinates
transforming non-hydrodynamic evolution of waterbag reduction to non-
homogeneous Riemann invariants form of hydrodynamic type.

Keywords: Manakov-Santini hierarchy, Lax representation, Waterbag reduction,
Non-homogeneous systems of hydrodynamic type, Riemann invariants
PACS: 02.30.Ik

##  1 Introduction

In this paper we study an integrable system introduced recently by Manakov and
Santini [1] (see also [2, 3]). This system is connected with commutation of
general 2-dimensional vector fields (containing derivative on spectral
variable). Reduction to Hamiltonian vector fields leads to the well-known
dispersionless KP (or Khokhlov-Zabolotskaya) equation. Alternatively, a
natural reduction to 1-dimensional vector fields reduces Manakov-Santini
system to the equation introduced by Pavlov [4] (see also [5, 6, 7]). Using
general construction of the works [8, 11], we introduce the hierarchy for
Manakov-Santini system in Lax-Sato form and generating equation for it (the
hierarchy in terms of recursion operator was introduced in [2]). We introduce
waterbag ansatz for Manakov-Santini hierarchy and derive equations of the
reduced hierarchy. Using rational form of the G function (see below), one can
introduce new coordinates such that the non-hydrodynamic evolution of waterbag
reduction transforms to non-homogeneous Riemann invariants form of hydro-
dynamic type.
This paper is organized as follows. In section 2, GdKP hierarchy is described,
connection to Manakov-Santini system is demonstrated. In section 3, waterbag
reduction for Manakov-Santini hierarchy is introduced, equations of reduced
hierarchy are derived (in non-hydrodynamic form). In section 4, we introduce
new coordinates transforming the evolution of waterbag reduction to non-
homogeneous Riemann invariants form of hydro-dynamic type. The examples are
given. Section 5 is devoted to the concluding remarks.

##  2 Generalized dKP hierarchy

To introduce generalized dKP (Manakov-Santini) hierarchy, we use general
construction of the works [8, 11]. The hierarchy is described by the Lax-Sato
equations

$\displaystyle\frac{\partial\psi}{\partial
t_{n}}=A_{n}\frac{\partial\psi}{\partial x}-B_{n}\frac{\partial\psi}{\partial
p},\qquad\psi=\left(\begin{array}[]{c}\mathcal{L}\\\
\mathcal{M}\end{array}\right),$ (3)

or, equivalently, by the generating equation

$(J_{0}^{-1}\mathrm{d}\mathcal{L}\wedge\mathrm{d}\mathcal{M})_{-}=0,$ (4)

where $A_{n}\equiv(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial p)_{+}$,
$B_{n}\equiv(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial x)_{+}$ with the Lax
and Orlov operators $\mathcal{L}(p),\mathcal{M}(p)$ being the Laurent series

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\sum_{n=1}^{\infty}u_{n}(x)p^{-n},$ (5) $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\sum_{n=1}^{\infty}v_{n}(x)\mathcal{L}^{-n}.$
(6)

Here $(\cdots)_{+}$ ($(\cdots)_{-}$) denote respectively the projection on the
polynomial part (negative powers), and $J_{0}$ is defined by

$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\frac{\partial\mathcal{M}}{\partial x}-\frac{\partial\mathcal{L}}{\partial
x}\frac{\partial\mathcal{M}}{\partial p}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\left(\left.\frac{\partial\mathcal{M}}{\partial\mathcal{L}}\right|_{t_{n},v_{n}\
\rm{fixed}}\frac{\partial\mathcal{L}}{\partial
x}+\left.\frac{\partial\mathcal{M}}{\partial x}\right|_{\mathcal{L}\
\rm{fixed}}\right)-\frac{\partial\mathcal{L}}{\partial
x}\left.\frac{\partial\mathcal{M}}{\partial\mathcal{L}}\right|_{t_{n},v_{n}\
\rm{fixed}}\frac{\partial\mathcal{L}}{\partial p}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial
p}\left.\frac{\partial\mathcal{M}}{\partial x}\right|_{\mathcal{L}\
\rm{fixed}}=1+v_{1x}p^{-1}+(v_{2x}-u_{1})p^{-2}+\cdots.$

We list some of $A_{n}$ and $B_{n}$ as follows

$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
A_{2}$ $\displaystyle=$ $\displaystyle 2p-2v_{1x},$ $\displaystyle A_{3}$
$\displaystyle=$ $\displaystyle 3p^{2}-3v_{1x}p+6u_{1}+3(v_{1x})^{2}-3v_{2x},$
$\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle
4p^{3}-4v_{1x}p^{2}+(12u_{1}+4(v_{1x})^{2}-4v_{2x})p$ (7)
$\displaystyle+12u_{2}-4v_{3x}+8v_{1x}v_{2x}-4(v_{1x})^{3}-8u_{1}v_{1x},$

and

$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B_{2}$ $\displaystyle=$ $\displaystyle 2u_{1x},$ $\displaystyle B_{3}$
$\displaystyle=$ $\displaystyle 3u_{1x}p-3u_{1x}v_{1x}+3u_{2x},$
$\displaystyle B_{4}$ $\displaystyle=$ $\displaystyle
4u_{1x}p^{2}+(4u_{2x}-4u_{1x}v_{1x})p$ (8)
$\displaystyle+4u_{1x}(4u_{1}+(v_{1x})^{2}-v_{2x})-4u_{2x}v_{1x}+4u_{3x}.$

The $t_{1}$ flow of the generalized dKP hierarchy (3) says that the dependence
on $t_{1}$ and $x$ appear in the linear combination $t_{1}+x$.

###### Proposition 2.1

The compatibility of the commuting flow
$[\partial_{t_{m}},\partial_{t_{n}}]\psi=0$ requires $A_{n},B_{n}$ to satisfy

$\displaystyle\partial_{t_{m}}A_{n}-\partial_{t_{n}}A_{m}$ $\displaystyle=$
$\displaystyle\langle A_{m},A_{n}\rangle_{x}+B_{n}A_{mp}-B_{m}A_{np},$
$\displaystyle\partial_{t_{m}}B_{n}-\partial_{t_{n}}B_{m}$ $\displaystyle=$
$\displaystyle\langle B_{n},B_{m}\rangle_{p}+A_{m}B_{nx}-A_{n}B_{mx},$ (9)

where $\langle U,V\rangle_{i}:=U(\partial_{i}V)-(\partial_{i}U)V$.

Proof. Substituting (3) into
$\partial_{t_{m}}\partial_{t_{n}}\psi=\partial_{t_{n}}\partial_{t_{m}}\psi$,
and comparing the coefficients of independent variables $\psi_{x}$ and
$\psi_{p}$ respectively to the both sides, we obtain (9). $\square$

The evolution of $\mathcal{L},\mathcal{M}$ with respect to $t_{2}=y$ in (3)
are given by

$\displaystyle\frac{1}{2}\frac{\partial\mathcal{L}}{\partial y}$
$\displaystyle=$ $\displaystyle(p-v_{1x})\frac{\partial\mathcal{L}}{\partial
x}-u_{1x}\frac{\partial\mathcal{L}}{\partial p},$ (10)
$\displaystyle\frac{1}{2}\frac{\partial\mathcal{M}}{\partial y}$
$\displaystyle=$ $\displaystyle(p-v_{1x})\frac{\partial\mathcal{M}}{\partial
x}-u_{1x}\frac{\partial\mathcal{M}}{\partial p}.$ (11)

Using the convention $(\sum_{n}a_{n}p^{n})_{[s]}=a_{s}$ for a formal Laurent
series, then from Eq.(10) we have

$\displaystyle\frac{1}{2}u_{1y}$ $\displaystyle=$
$\displaystyle\Big{(}(p-v_{1x})\mathcal{L}_{x}-u_{1x}\mathcal{L}_{p}\Big{)}_{[-1]}=u_{2x}-v_{1x}u_{1x},$
(12) $\displaystyle\frac{1}{2}u_{2y}$ $\displaystyle=$
$\displaystyle\Big{(}(p-v_{1x})\mathcal{L}_{x}-u_{1x}\mathcal{L}_{p}\Big{)}_{[-2]}=u_{3x}-v_{1x}u_{2x}+u_{1}u_{1x},$
(13)

On the other hand, the expression of Eq.(11) together with (10) gives

$\mathcal{L}+\frac{1}{2}\sum_{n=1}^{\infty}v_{ny}\mathcal{L}^{-n}=(p-v_{1x})\left(1+\sum_{n=1}^{\infty}v_{nx}\mathcal{L}^{-n}\right).$

Comparing the coefficients of powers $p^{-1}$ and $p^{-2}$ to the above, we
have

$\displaystyle v_{2x}$ $\displaystyle=$ $\displaystyle
u_{1}+v_{1x}^{2}+\frac{1}{2}v_{1y},$ (14) $\displaystyle v_{3x}$
$\displaystyle=$ $\displaystyle
u_{2}+\frac{1}{2}v_{2y}+u_{1}v_{1x}+v_{1x}v_{2x}.$ (15)

Similarly, the evolution of $\mathcal{L},\mathcal{M}$ w.r.t. $t_{3}=t$ are
given by

$\displaystyle\frac{1}{3}\frac{\partial\mathcal{L}}{\partial t}$
$\displaystyle=$
$\displaystyle\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\frac{\partial\mathcal{L}}{\partial
x}-\left(u_{1x}p+\frac{1}{2}u_{1y}\right)\frac{\partial\mathcal{L}}{\partial
p},$ (16) $\displaystyle\frac{1}{3}\frac{\partial\mathcal{M}}{\partial t}$
$\displaystyle=$
$\displaystyle\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\frac{\partial\mathcal{M}}{\partial
x}-\left(u_{1x}p+\frac{1}{2}u_{1y}\right)\frac{\partial\mathcal{M}}{\partial
p},$ (17)

Then the $t$-flow of $u_{1}$ can be read by Eq.(16) by taking the coefficient
of $p^{-1}$:

$\displaystyle\frac{1}{3}u_{1t}$ $\displaystyle=$ $\displaystyle
u_{3x}-v_{1x}u_{2x}+(u_{1}-\frac{1}{2}v_{1y})u_{1x}+u_{1}u_{1x},$ (18)
$\displaystyle=$
$\displaystyle\frac{1}{2}u_{2y}-\frac{1}{2}u_{1x}v_{1y}+u_{1}u_{1x},$

where we have used (13) to reach the second line. Also, the expression of
Eq.(17) together with (16) gives

$\mathcal{L}^{2}+\frac{1}{3}\sum_{n=1}^{\infty}v_{nt}\mathcal{L}^{-n}=\left(p^{2}-v_{1x}p+u_{1}-\frac{1}{2}v_{1y}\right)\left(1+\sum_{n=1}^{\infty}v_{nx}\mathcal{L}^{-n}\right),$

in which the coefficient of $p^{-1}$ gives

$\frac{1}{3}v_{1t}=-u_{2}+\frac{1}{2}v_{2y}+u_{1}v_{1x}-\frac{1}{2}v_{1x}v_{1y},$
(19)

where we have used Eq.(15). Now differentiating Eqs.(18), (19) respectively
with respect to $x$ and eliminating $u_{2x}$ and $v_{2x}$ by Eqs.(12) and
(14), we obtain the following two coupled equations for $u_{1}:=u$ and
$v_{1}=v$:

$\displaystyle\frac{1}{3}u_{xt}$ $\displaystyle=$
$\displaystyle\frac{1}{4}u_{yy}+(uu_{x})_{x}+\frac{1}{2}v_{x}u_{xy}-\frac{1}{2}u_{xx}v_{y},$
$\displaystyle\frac{1}{3}v_{xt}$ $\displaystyle=$
$\displaystyle\frac{1}{4}v_{yy}+uv_{xx}+\frac{1}{2}v_{x}v_{xy}-\frac{1}{2}v_{xx}v_{y}.$
(20)

Eq.(20) is the so called _Manakov-Santini equation_ [1, 2, 3]. The Lax pair
for this equation is defined by linear equations (10,11) and (16,17). Notice
that for $v=0$ reduction, the system reduces to the dKP equation

$\frac{1}{3}u_{xt}=\frac{1}{4}u_{yy}+(uu_{x})_{x}.$ (21)

Respectively, $u=0$ reduction gives an equation [4] (see also [5, 6, 7])

$\frac{1}{3}v_{xt}=\frac{1}{4}v_{yy}+\frac{1}{2}v_{x}v_{xy}-\frac{1}{2}v_{xx}v_{y}.$
(22)

###### Proposition 2.2

Equation (3) can be written in Hamilton-Jacobi type equation

$\left.\frac{\partial p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}\
\rm{fixed}}=\left.A_{n}(p(\mathcal{L}))\frac{\partial p(\mathcal{L})}{\partial
x}\right|_{\mathcal{L}\ \rm{fixed}}+B_{n}(p(\mathcal{L})),$ (23)

where $A_{n}(p)=(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial p)_{+}$ and
$B_{n}(p)=(J_{0}^{-1}\partial\mathcal{L}^{n}/\partial x)_{+}$.

Proof. By taking into account the partial derivatives with respect to $t_{n}$
for fixed $p$ or $\mathcal{L}$, it is easy to show that

$\frac{\partial p}{\partial t_{n}}=0=\left.\frac{\partial
p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}}+\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
t_{n}},$

or

$\left.\frac{\partial p(\mathcal{L})}{\partial
t_{n}}\right|_{\mathcal{L}}=-\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\frac{\partial\mathcal{L}}{\partial
t_{n}}.$ (24)

Using (3), and (24) with $n=1$, we have

$\left.\frac{\partial p(\mathcal{L})}{\partial t_{n}}\right|_{\mathcal{L}\
\rm{fixed}}=-\frac{\partial
p(\mathcal{L})}{\partial\mathcal{L}}\left(A_{n}(p)\frac{\partial\mathcal{L}}{\partial
x}-B_{n}(p)\frac{\partial\mathcal{L}}{\partial
p}\right)=\left.A_{n}(p(\mathcal{L}))\frac{\partial p(\mathcal{L})}{\partial
x}\right|_{\mathcal{L}\ \rm{fixed}}+B_{n}(p(\mathcal{L})).\,\square$

###### Proposition 2.3

The function
$J_{0}=\partial_{p}\mathcal{L}\partial_{x}\mathcal{M}-\partial_{x}\mathcal{L}\partial_{p}\mathcal{M}$
and its inverse $G=J_{0}^{-1}$ satisfy

$\displaystyle\partial_{t_{n}}J_{0}$ $\displaystyle=$
$\displaystyle\left(A_{n}J_{0}\right)_{x}-\left(B_{n}J_{0}\right)_{p},$ (25)
$\displaystyle\partial_{t_{n}}G$ $\displaystyle=$ $\displaystyle\langle
A_{n},G\rangle_{x}-\langle B_{n},G\rangle_{p},$ (26)

where $\langle U,V\rangle_{i}:=U(\partial_{i}V)-(\partial_{i}U)V$.

Proof. Using the $t_{n}$-flows of $\mathcal{L},\mathcal{M}$ in (3) and the
definition of $J_{0}$, we have

$\displaystyle\partial_{t_{n}}J_{0}$ $\displaystyle=$
$\displaystyle(\mathcal{L}_{p})_{t_{n}}\mathcal{M}_{x}+\mathcal{L}_{p}(\mathcal{M}_{x})_{t_{n}}-(\mathcal{L}_{x})_{t_{n}}\mathcal{M}_{p}-\mathcal{L}_{x}(\mathcal{M}_{p})_{t_{n}},$
$\displaystyle=$ $\displaystyle-
B_{np}J_{0}+A_{nx}J_{0}+A_{n}J_{0x}-B_{n}J_{0p},$ $\displaystyle=$
$\displaystyle(A_{n}J_{0})_{x}-(B_{n}J_{0})_{p}.$

Moreover, substituting $J_{0}=G^{-1}$ into the above we obtain (26). $\square$
As we will see, Proposition 2.3 can provide a crucial way to determine the
hierarchy flows.

##  3 Waterbag-type reduction

Consider the waterbag-type reduction of the generalized dKP hierarchy
represented by [8]

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\sum_{i=1}^{N}\epsilon_{i}\log(p-U_{i}),$ (27) $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\sum_{i=1}^{M}\delta_{i}\log(p-V_{i}),$
(28)

where $\epsilon_{i}$ and $\delta_{i}$ are assumed to satisfy

$\sum_{i=1}^{N}\epsilon_{i}=\sum_{i=1}^{M}\delta_{i}=0.$ (29)

The ansatz (27,28) is consistent with the dynamics defined by Manakov-Santini
hierarchy (3), i.e., the form of ansatz is preserved by the dynamics.
Condition (29) guarantees that expansion of $\mathcal{L}$, $\mathcal{M}$ at
infinity is of the form (5,6). Reduced hierarchy is represented as infinite
set of (1+1)-dimensional systems of equations for the functions $U_{i}$,
$V_{i}$, which are obtained by the substitution of ansatz (27,28) to equations
of Manakov-Santini hierarchy (3).

Let us consider first flows of reduced hierarchy. For expansion of
$\mathcal{L}$, $\mathcal{M}$ at infinity from (27,28) we get

$\displaystyle\mathcal{L}=p-\sum_{n=1}^{\infty}\left(\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{n}}{n}\right)p^{-n},$
(30)
$\displaystyle\mathcal{M}=\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}-\sum_{n=1}^{\infty}\left(\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{n}}{n}\right)p^{-n}.$
(31)

Comparing these expansions with formulae (5,6), we come to the conclusion that
$u_{n}=-\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{n}}{n}$. To calculate $v_{n}$,
we should invert the series (30) to find $p(\mathcal{L})$ that can be done
recursively, and substitute $p(\mathcal{L})$ to (31) . For the first
coefficients $u_{n}$, $v_{n}$ we get

$\displaystyle u_{1}=-\sum_{i=1}^{N}\epsilon_{i}U_{i},\quad
u_{2}=-\frac{1}{2}\sum_{i=1}^{N}\epsilon_{i}U_{i}^{2},$ $\displaystyle
v_{1}=-\sum_{i=1}^{M}\delta_{i}V_{i},\quad
v_{2}=-\frac{1}{2}\sum_{i=1}^{M}\delta_{i}V_{i}^{2}.$

Substituting these expressions to relations (7), (8) and using equations (3),
we obtain equations of reduced hierarchy. Equations of the flow corresponding
to $y=t_{2}$ read

$\displaystyle\partial_{y}U_{k}=\Big{(}2U_{k}+\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\Big{)}\partial_{x}U_{k}-2\partial_{x}\Big{(}\sum_{i=1}^{N}\epsilon_{i}U_{i}\Big{)},$
$\displaystyle\partial_{y}V_{k}=\Big{(}2V_{k}+\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\Big{)}\partial_{x}V_{k}-2\partial_{x}\Big{(}\sum_{i=1}^{N}\epsilon_{i}U_{i}\Big{)}.$
(32)

For the flow corresponding to $t=t_{3}$ we get

$\displaystyle\partial_{t}U_{k}=\left(3U_{k}^{2}+3U_{k}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}-6\sum_{i=1}^{N}\epsilon_{i}U_{i}+3\bigl{(}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\bigr{)}^{2}+3\partial_{x}\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{2}}{2}\right)\partial_{x}U_{k}-$
$\displaystyle\qquad\left(3U_{k}\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i}+3(\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i})(\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i})+3\partial_{x}\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{2}}{2}\right),$
$\displaystyle\partial_{t}V_{k}=\left(3V_{k}^{2}+3V_{k}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}-6\sum_{i=1}^{N}\epsilon_{i}U_{i}+3\bigl{(}\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i}\bigr{)}^{2}+3\partial_{x}\sum_{i=1}^{M}\delta_{i}\frac{V_{i}^{2}}{2}\right)\partial_{x}V_{k}-$
$\displaystyle\qquad\left(3V_{k}\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i}+3(\partial_{x}\sum_{i=1}^{N}\epsilon_{i}U_{i})(\partial_{x}\sum_{i=1}^{M}\delta_{i}V_{i})+3\partial_{x}\sum_{i=1}^{N}\epsilon_{i}\frac{U_{i}^{2}}{2}\right).$
(33)

A common solution to the systems (32), (33) gives a solution to Manakov-
Santini equation (20) defined as $u=-\sum_{i=1}^{N}\epsilon_{i}U_{i}$,
$v=-\sum_{i=1}^{M}\delta_{i}V_{i}$.

##  4 Diagonal form of reduced hierarchy

For the waterbag reduction (27, 28) one can show that the $G$ function can be
expressed in the following form

$G=J_{0}^{-1}=\frac{\prod_{i=1}^{N}(p-U_{i})\prod_{j=1}^{M}(p-V_{j})}{F(U_{n},U_{nx},V_{m},V_{mx};p)},\qquad
n=1,\ldots,N;\,m=1,\ldots,M,$ (34)

where the function $F$ in denominator is a polynomial of $p$ with degree
$N+M$. In general, $F$ can also be factorized into
$\prod_{k=1}^{N+M}(p-W_{k})$, for which
$W_{k}=W_{k}(U_{n},U_{nx},V_{m},V_{mx})$ are roots of $J_{0}$. We like to
mention here that the derivatives $U_{nx},V_{mx}$ can be inversely expressed
as function of the form

$U_{nx}=f_{n}(U_{i},V_{j},W_{k}),\quad V_{mx}=g_{m}(U_{i},V_{j},W_{k}).$ (35)

Therefore, we have

$J_{0}=\frac{\prod_{k=1}^{N+M}(p-W_{k})}{\prod_{i=1}^{N}(p-U_{i})\prod_{j=1}^{M}(p-V_{j})},\qquad
n=1,\ldots,N;\,m=1,\ldots,M.$ (36)

As the result, the evaluation of $G$ at $U_{i}$ or $V_{i}$, i.e.,
$G(p=U_{i})=0$ or $G(p=V_{i})=0$ shows that Eq.(26) can be written into the
following evolution equations of $U_{i}$, $V_{i}$:

$\displaystyle\frac{\partial U_{i}}{\partial t_{n}}$ $\displaystyle=$
$\displaystyle A_{n}(p=U_{i})\frac{\partial U_{i}}{\partial
x}+B_{n}(p=U_{i}),$ (37) $\displaystyle\frac{\partial V_{i}}{\partial t_{n}}$
$\displaystyle=$ $\displaystyle A_{n}(p=V_{i})\frac{\partial V_{i}}{\partial
x}+B_{n}(p=V_{i}).$ (38)

Similarly, Eq.(25) with $J_{0}(p=W_{i})=0$ gives rise

$\frac{\partial W_{i}}{\partial t_{n}}=A_{n}(p=W_{i})\frac{\partial
W_{i}}{\partial x}+B_{n}(p=W_{i}).$ (39)

In summary, combining (37), (38), (39) and replacing those $U_{nx}$’s and
$V_{mx}$’s in $A_{n},B_{n}$ with the transformations (35), we obtain the _non-
homogeneous Riemann invariant form_ as

$\partial_{t_{n}}R_{i}=A_{n}(p=R_{i})R_{ix}+B_{n}(p=R_{i}),\quad
i=1,\ldots,2N+2M,$ (40)

for which
$(R_{1},\ldots,R_{2N+2M})=(U_{1},\ldots,U_{N};V_{1},\ldots,V_{M};W_{1},\ldots,W_{N+M})$.
Some linearly degenerate non-homogeneous Riemann invariants forms, associated
with commuting quadratic Hamiltonians and the Killing vector fields of the
given metric, were investigated in [9, 10]. However, in our case equation (40)
is obviously not linearly degenerate.

Remark. For the type of non-homogeneous Riemann invariant form

$\partial_{t_{n}}R^{i}=\Lambda_{n}^{i}(\mathbf{R})R^{i}_{x}+Q_{n}^{i}(\mathbf{R}),$
(41)

the requirements of the commutativity are equivalent to the following
restrictions on their characteristic speeds and non-homogeneous terms (see
appendix A)

$\frac{\partial_{j}\Lambda_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}\Lambda_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}},\quad\frac{\partial_{j}Q_{n}^{i}}{Q_{n}^{j}}=\frac{\partial_{j}Q_{m}^{i}}{Q_{m}^{j}},\quad\frac{Q_{n}^{j}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{Q_{m}^{j}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}},\quad
i\neq j,\quad n\neq m.$

where $\partial_{i}\equiv\partial/\partial R^{i}$.
Example 1. $(N,M)=(1,1)$ reduction. In this case,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle p+\log(1-U/p),$
$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\log(1-V/p).$

Comparing to the expansion of (5,6) we have $u_{n}=-U^{n}/n$ for $n\geq 1$ and
$v_{1}=-V,v_{2}=-V^{2}/2,v_{3}=UV-V^{3}/3$, etc. These transformations allow
us to get $A_{n},B_{n}$ (by Eqs.(7), (8)) which correspond to the reduced
system. The $G$ function is given by

$G=\frac{p(p-U)(p-V)}{\prod_{i=1}^{3}(p-W_{i})},$ (42)

where $W_{i}$ satisfy

$\sum_{i=1}^{3}W_{i}=U+V+V_{x},\quad\sum_{\scriptstyle
i,j=1\atop\scriptstyle(i>j)}^{3}W_{i}W_{j}=U+UV+UV_{x},\quad\prod_{i=1}^{3}W_{i}=UV+UV_{x}-U_{x}V.$
(43)

Notice that (42) is not coincident with that in (34), there is one more root
of $p=0$ to be considered. By (26), it turns out that the evaluation of $p=0$
gives an additional condition, namely

$UVB_{n}(p=0)=0,\quad\forall n\geq 1.$ (44)

There are two simple cases: (i) $V=0,U\neq 0$, (ii) $V\neq 0,U=0$. One can
easily deduce considering $t_{2}$-flow of (40) that case (i) is a trivial
reduction. For the case (ii), we have the fact that $B_{n}(U=0)=0$ for $n\geq
1$, and Eq.(43) will reveal us the only one relation: $V_{x}=W-V$. To this
end, system (40) reduces to the type of homogeneous one in (41) with
$Q_{n}^{i}=0$, namely

$\partial_{t_{n}}R^{i}=\Lambda_{n}^{i}(\mathbf{R})\partial_{x}R^{i},$ (45)

where $\mathbf{R}=(R^{1},R^{2})=(V,W)$ and the characteristic speeds
$\Lambda_{n}^{i}=A_{n}(p=R^{i},U=0)$. For instance, for $t_{2}=y$ flow, we
have $A_{2}(U=0)=2p+2V_{x}=2p+2(W-V)$, then Eq.(45) becomes

$\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{y}=\left(\begin{array}[]{cc}2W&0\\\
0&4W-2V\end{array}\right)\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{x}.$ (46)

For $t_{3}=t$ flow, we derive $A_{3}(U=0)=3p^{2}+3(W-V)p+3(W-V)^{2}+3V(W-V)$,
thus

$\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{t}=\left(\begin{array}[]{cc}3W^{2}&0\\\
0&9W^{2}-6VW\end{array}\right)\left(\begin{array}[]{c}V\\\
W\end{array}\right)_{x}.$ (47)

From the two nontrivial flows (46), (47), we readily obtain the following set
of hodograph equation

$\displaystyle x+2Wy+3W^{2}t$ $\displaystyle=$ $\displaystyle\hat{F}(V,W),$
$\displaystyle x+(4W-2V)y+(9W^{2}-6VW)t$ $\displaystyle=$
$\displaystyle\hat{G}(V,W),$ (48)

where $\hat{F}$ and $\hat{G}$ satisfy the linear equations

$\displaystyle(W-V)\hat{G}_{V}$ $\displaystyle=$
$\displaystyle\hat{F}-\hat{G},$ $\displaystyle(W-V)\hat{F}_{W}$
$\displaystyle=$ $\displaystyle\hat{G}-\hat{F}.$

Dividing these two equations for $V\neq W$ we get $\hat{G}_{V}=-\hat{F}_{W}$.
It follows that there exists a function $\phi$ such that
$\hat{F}=\phi_{V},\hat{G}=-\phi_{W}$, whence $\phi$ satisfies the defining
equation

$(V-W)\phi_{VW}=\phi_{V}+\phi_{W}.$ (49)

Eq. (49) has a general solution of the form

$\phi=(V-W)\left(f(W)+\int\frac{g(V)}{(V-W)^{2}}\,dV\right),$

where $f(W)$ and $g(V)$ are arbitrary functions of $W$ and $V$, respectively.
Choosing, for example $f(W)=W^{3}$, $g(V)=\mbox{Const.}$, then we have
$\hat{F}=W^{3}$ and $\hat{G}=-3VW^{2}+4W^{3}$. Substituting back into the
hodograph equation (48) we solve

$\displaystyle V=W$ $\displaystyle=$
$\displaystyle\frac{1}{6}h^{-1/3}(24y+36t^{2}+6th^{1/3}+h^{2/3}),$
$\displaystyle h$ $\displaystyle=$ $\displaystyle
216yt+108x+216t^{3}+12\sqrt{-96y^{3}-108y^{2}t^{2}+324ytx+81x^{2}+324xt^{3}},$

which satisfies the $t_{2}$\- and $t_{3}$-flows (46), (47). However, $V=W$
contradicts to the relation $W=V+V_{x}$ and $V$ does not satisfy equation
(22). Actually, from equation (49) we can see that when $V=W$ one can get
$\hat{F}=\hat{G}$. Then we obtain all the solutions will satisfy $V=W$.
Consequently, there is no (1,1)- reduction. Similar considerations can show
that there are no (1,2)- and (2,1)- reductions, either.
Example 2. $(N,M)=(2,2)$ reduction. In this case,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
p+\epsilon_{1}\log\frac{p-U_{1}}{p-U_{2}},$ $\displaystyle\mathcal{M}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}nt_{n}\mathcal{L}^{n-1}+\delta_{1}\log\frac{p-V_{1}}{p-V_{2}}.$

For simplicity, we set $\epsilon_{1}=\delta_{1}=1$. Comparing to the expansion
of (5,6), we have $u_{n}=(U_{2}^{n}-U_{1}^{n})/n$ for $n\geq 1$ and
$v_{1}=V_{2}-V_{1},v_{2}=(V_{2}^{2}-V_{1}^{2})/2,v_{3}=u_{1}v_{1}+(V_{2}^{3}-V_{1}^{3})/3,\ldots$.
Now we expand the hierarchy flow of $U_{i}$, $V_{i}$ and $W_{i}$ up to
$t_{2}=y,t_{3}=t$. From (34) with $(N,M)=(2,2)$ we have

$G=\frac{\prod_{i=1}^{2}(p-U_{i})\prod_{j=1}^{2}(p-V_{j})}{\prod_{k=1}^{4}(p-W_{k})}.$

where $W_{i}$ satisfy

$\displaystyle\sum_{i=1}^{4}W_{i}$ $\displaystyle=$ $\displaystyle
U_{1}+U_{2}+V_{1}+V_{2}+V_{1x}-V_{2x},$ (50) $\displaystyle\sum_{\scriptstyle
i,j=1\atop\scriptstyle(i>j)}^{4}W_{i}W_{j}$ $\displaystyle=$ $\displaystyle
U_{1}-U_{2}+U_{1}U_{2}+V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x}$ (51)
$\displaystyle+(U_{1}+U_{2})(V_{1}+V_{2}+V_{1x}-V_{2x}),$
$\displaystyle\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}$ $\displaystyle=$
$\displaystyle(U_{1}+U_{2})(V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x})+(U_{2x}-U_{1x})(V_{1}-V_{2})$
(52) $\displaystyle+(U_{1}-U_{2}+U_{1}U_{2})(V_{1}+V_{2}+V_{1x}-V_{2x}),$
$\displaystyle\prod_{i=1}^{4}W_{i}$ $\displaystyle=$
$\displaystyle(U_{1}-U_{2}+U_{1}U_{2})(V_{1}V_{2}+V_{1x}V_{2}-V_{1}V_{2x})$
(53) $\displaystyle-(V_{1}-V_{2})(U_{1x}U_{2}-U_{1}U_{2x}),$

from which, one can substitute into $A_{n},B_{n}$ to eliminate
$U_{ix},V_{ix}$, etc. For $n=2$, using (50), (52) we have

$\displaystyle A_{2}(p)$ $\displaystyle=$ $\displaystyle
2p+2(V_{1x}-V_{2x})=2p+2\Big{(}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)},$
$\displaystyle=$ $\displaystyle
2(p-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8}),$

and the non-homogeneous term

$\displaystyle B_{2}(p)=2(U_{2x}-U_{1x}),$
$\displaystyle=\frac{2}{V_{1}-V_{2}}\Bigg{[}\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}+(U_{1}+U_{2})\bigg{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}^{4}W_{i}W_{j}\bigg{)}$
$\displaystyle\quad+\Bigg{.}\bigg{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\bigg{)}\bigg{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\bigg{)}\Bigg{]},$
$\displaystyle=\frac{2}{R_{3}-R_{4}}\Big{[}(R_{1}+R_{2}-R_{5}-R_{6}-R_{7}-R_{8})(R_{1}-R_{2}-R_{1}R_{2}-R_{1}^{2}-R_{2}^{2})\Big{.}$
$\displaystyle\quad+(R_{1}+R_{2})(R_{1}-R_{2}+R_{1}R_{2}-R_{5}R_{6}-R_{5}R_{7}-R_{5}R_{8}-R_{6}R_{7}-R_{6}R_{8}-R_{7}R_{8})$
$\displaystyle\quad+R_{5}R_{6}R_{7}+R_{6}R_{7}R_{8}+R_{7}R_{8}R_{5}+R_{8}R_{5}R_{6}\Big{]}.$

Then the $t_{2}=y$ flow in (40) is now read

$\partial_{y}R_{i}=2(R_{i}-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8})R_{ix}+B_{2}.$
(54)

For $n=3$, Eq.(40) becomes

$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle A_{3}(p=R_{i})R_{ix}+B_{3}(p=R_{i}),$ $\displaystyle=$
$\displaystyle\left[3p^{2}+3(V_{1x}-V_{2x})p+6(U_{2}-U_{1})+3(V_{1x}-V_{2x})^{2}+\frac{3}{2}(V_{1}^{2}-V_{2}^{2})_{x}\right]_{p=R_{i}}R_{ix}$
$\displaystyle+\left[3(U_{2x}-U_{1x})p-3(U_{2x}-U_{1x})(V_{2x}-V_{1x})+\frac{3}{2}(U_{2}^{2}-U_{1}^{2})_{x}\right]_{p=R_{i}}$
$\displaystyle=$
$\displaystyle\left[3R_{i}^{2}+3R_{i}(V_{1x}-V_{2x})+6(U_{2}-U_{1})+3(V_{1x}-V_{2x})^{2}+\frac{3}{2}(V_{1}^{2}-V_{2}^{2})_{x}\right]R_{ix}$
$\displaystyle+3R_{i}(U_{2x}-U_{1x})-3(U_{2x}-U_{1x})(V_{2x}-V_{1x})+3(U_{2}U_{2x}-U_{1}U_{1x}).$

Using Eqs.(50)–(53) we arrive

$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle
3R_{ix}\Bigg{[}U_{2}-U_{1}+R_{i}\Big{(}R_{i}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)}$
$\displaystyle+U_{1}U_{2}-V_{1}V_{2}-V_{1}^{2}-V_{2}^{2}-\sum_{i>j}^{4}W_{i}W_{j}+\Big{(}U_{1}+U_{2}+V_{1}+V_{2}-\sum_{i=1}^{4}W_{i}\Big{)}^{2}$
$\displaystyle\qquad+(U_{1}+U_{2}+V_{1}+V_{2})\Big{(}-U_{1}-U_{2}+\sum_{i=1}^{4}W_{i}\Big{)}\Bigg{]}$
$\displaystyle+\frac{3R_{i}}{V_{1}-V_{2}}\Bigg{[}(U_{1}+U_{2})\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}^{4}W_{i}W_{j}\Big{)}+\sum_{i=1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}$
$\displaystyle\quad+\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}\Bigg{]}+$
$\displaystyle+3\Big{(}-U_{1}-U_{2}-V_{1}-V_{2}+\sum_{i=1}^{4}W_{i}\Big{)}\times$
$\displaystyle\times\frac{1}{V_{1}-V_{2}}\Bigg{(}\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}$
$\displaystyle\qquad+(U_{1}+U_{2})\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}W_{i}W_{j}\Big{)}+\sum_{i-1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}\Bigg{)}$
$\displaystyle+\frac{3}{V_{1}-V_{2}}\Bigg{(}(U_{1}+U_{2})\sum_{i-1}^{4}W_{i}^{-1}\prod_{j=1}^{4}W_{j}+(U_{1}+U_{2})^{2}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-\sum_{i>j}W_{i}W_{j}\Big{)}$
$\displaystyle-\prod_{i=1}^{4}W_{i}+(U_{1}+U_{2})\Big{(}U_{1}+U_{2}-\sum_{i=1}^{4}W_{i}\Big{)}\Big{(}U_{1}-U_{2}+U_{1}U_{2}-(U_{1}+U_{2})^{2}\Big{)}$
$\displaystyle+(U_{1}-U_{2}+U_{1}U_{2})\Big{(}\sum_{i>j}^{4}W_{i}W_{j}-(U_{1}-U_{2})-U_{1}U_{2}+(U_{1}+U_{2})^{2}-(U_{1}+U_{2})\sum_{i=1}^{4}W_{i}\Big{)}\Bigg{)}.$

Expressing in terms of $R_{i},i=1,\ldots,8$, we get

$\displaystyle\frac{\partial R_{i}}{\partial t}$ $\displaystyle=$
$\displaystyle
3R_{ix}\Bigg{[}R_{2}-R_{1}+R_{i}(R_{i}-R_{1}-R_{2}-R_{3}-R_{4}+R_{5}+R_{6}+R_{7}+R_{8})$
$\displaystyle\quad+R_{1}R_{2}+R_{3}R_{4}+R_{5}R_{6}+R_{5}R_{7}+R_{5}R_{8}+R_{6}R_{7}+R_{6}R_{8}+R_{7}R_{8}$
$\displaystyle\quad+R_{1}R_{3}+R_{1}R_{4}-R_{1}R_{5}-R_{1}R_{6}-R_{1}R_{7}-R_{1}R_{8}+R_{2}R_{3}+R_{2}R_{4}$
$\displaystyle\quad-
R_{2}R_{5}-R_{2}R_{6}-R_{2}R_{7}-R_{2}R_{8}-R_{3}R_{5}-R_{3}R_{6}-R_{3}R_{7}-R_{3}R_{8}$
$\displaystyle\quad-
R_{4}R_{5}-R_{4}R_{6}-R_{4}R_{7}-R_{4}R_{8}+R_{5}^{2}+R_{6}^{2}+R_{7}^{2}+R_{8}^{2}\Bigg{]}$
$\displaystyle+\frac{3R_{i}}{R_{3}-R_{4}}\Big{(}2R_{1}^{2}-2R_{2}^{2}-R_{1}R_{5}-R_{1}R_{6}-R_{1}R_{7}-R_{1}R_{8}+R_{2}R_{5}+R_{2}R_{6}$
$\displaystyle\quad+R_{2}R_{7}+R_{2}R_{8}-R_{1}^{3}-R_{2}^{3}-R_{1}^{2}R_{2}+R_{1}^{2}R_{5}+R_{1}^{2}R_{6}+R_{1}^{2}R_{7}+R_{1}^{2}R_{8}$
$\displaystyle\quad-
R_{1}R_{2}^{2}+R_{2}^{2}R_{5}+R_{2}^{2}R_{6}+R_{2}^{2}R_{7}+R_{2}^{2}R_{8}+R_{1}R_{2}R_{5}+R_{1}R_{2}R_{6}$
$\displaystyle\quad+R_{1}R_{2}R_{7}+R_{1}R_{2}R_{8}-R_{1}R_{5}R_{6}-R_{1}R_{5}R_{7}-R_{1}R_{5}R_{8}-R_{1}R_{6}R_{7}$
$\displaystyle\quad-
R_{1}R_{6}R_{8}-R_{1}R_{7}R_{8}-R_{2}R_{5}R_{6}-R_{2}R_{5}R_{7}-R_{2}R_{5}R_{8}-R_{2}R_{6}R_{7}$
$\displaystyle\quad-
R_{2}R_{6}R_{8}-R_{2}R_{7}R_{8}+R_{5}R_{6}R_{7}+R_{6}R_{7}R_{8}+R_{7}R_{8}R_{5}+R_{8}R_{5}R_{6}\Big{)}+$
$\displaystyle+\frac{3}{R_{3}-R_{4}}\Big{(}-R_{1}^{2}-R_{2}^{2}+2R_{1}R_{2}+R_{1}^{3}-R_{2}^{3}-R_{1}^{2}R_{2}+R_{1}R_{2}^{2}-2R_{1}^{2}R_{3}+2R_{2}^{2}R_{3}$
$\displaystyle\quad-2R_{1}^{2}R_{4}+2R_{2}^{2}R_{4}+R_{1}^{2}R_{5}-R_{2}^{2}R_{5}+R_{1}^{2}R_{6}-R_{2}^{2}R_{6}+R_{1}^{2}R_{7}-R_{2}^{2}R_{7}$
$\displaystyle\quad+R_{1}^{2}R_{8}-R_{2}^{2}R_{8}-R_{1}R_{5}^{2}+R_{2}R_{5}^{2}+R_{1}R_{6}^{2}-R_{2}R_{6}^{2}+R_{1}R_{7}^{2}-R_{2}R_{7}^{2}$
$\displaystyle\quad+R_{1}R_{8}^{2}-R_{2}R_{8}^{2}+R_{1}^{3}R_{2}+R_{1}^{3}R_{3}+R_{1}^{3}R_{4}-R_{1}^{3}R_{5}-R_{1}^{3}R_{6}-R_{1}^{3}R_{7}-R_{1}^{3}R_{8}$
$\displaystyle\quad+R_{1}R_{2}^{3}+R_{2}^{3}R_{3}+R_{2}^{3}R_{4}-R_{2}^{3}R_{5}-R_{2}^{3}R_{6}-R_{2}^{3}R_{7}-R_{2}^{3}R_{8}$
$\displaystyle\quad+R_{1}^{2}R_{2}^{2}+R_{1}^{2}R_{5}^{2}+R_{2}^{2}R_{5}^{2}-R_{1}^{2}R_{6}^{2}-R_{2}^{2}R_{6}^{2}-R_{1}^{2}R_{7}^{2}-R_{2}^{2}R_{7}^{2}-R_{1}^{2}R_{8}^{2}-R_{2}^{2}R_{8}^{2}$
$\displaystyle\quad+R_{1}R_{5}R_{6}+R_{1}R_{5}R_{7}+R_{1}R_{5}R_{8}+3R_{1}R_{6}R_{7}+3R_{1}R_{6}R_{8}+3R_{1}R_{7}R_{8}$
$\displaystyle\quad-
R_{2}R_{5}R_{6}-R_{2}R_{5}R_{7}-R_{2}R_{5}R_{8}-3R_{2}R_{6}R_{7}-3R_{2}R_{6}R_{8}-3R_{2}R_{7}R_{8}$
$\displaystyle\quad+R_{3}R_{5}R_{1}-R_{3}R_{5}R_{2}-R_{3}R_{6}R_{1}+R_{3}R_{6}R_{2}-R_{3}R_{7}R_{1}+R_{3}R_{7}R_{2}$
$\displaystyle\quad-
R_{3}R_{8}R_{1}+R_{3}R_{8}R_{2}+R_{4}R_{5}R_{1}-R_{4}R_{5}R_{2}-R_{4}R_{6}R_{1}+R_{4}R_{6}R_{2}$
$\displaystyle\quad-
R_{4}R_{7}R_{1}+R_{4}R_{7}R_{2}-R_{4}R_{8}R_{1}+R_{4}R_{8}R_{2}-2R_{5}R_{1}^{2}R_{2}-2R_{6}R_{1}^{2}R_{2}$
$\displaystyle\quad-2R_{7}R_{1}^{2}R_{2}-2R_{1}R_{6}R_{2}^{2}-2R_{1}R_{7}R_{2}^{2}-2R_{1}R_{8}R_{2}^{2}-2R_{8}R_{1}^{2}R_{2}-2R_{1}^{2}R_{6}R_{7}$
$\displaystyle\quad-2R_{1}^{2}R_{6}R_{8}-2R_{1}^{2}R_{7}R_{8}-2R_{2}^{2}R_{6}R_{7}-2R_{2}^{2}R_{6}R_{8}-2R_{2}^{2}R_{7}R_{8}-2R_{1}R_{5}R_{2}^{2}$
$\displaystyle\quad+R_{5}^{2}R_{6}R_{7}+R_{7}R_{8}R_{5}^{2}+R_{8}R_{5}^{2}R_{6}+R_{5}^{2}R_{1}R_{2}-R_{1}R_{5}^{2}R_{6}-R_{1}R_{5}^{2}R_{7}-R_{1}R_{5}^{2}R_{8}$
$\displaystyle\quad-
R_{2}R_{5}^{2}R_{6}-R_{2}R_{5}^{2}R_{7}-R_{2}R_{5}^{2}R_{8}+R_{5}R_{6}^{2}R_{7}+R_{6}^{2}R_{7}R_{8}+R_{8}R_{5}R_{6}^{2}-R_{6}^{2}R_{1}R_{2}$
$\displaystyle\quad-
R_{1}R_{5}R_{6}^{2}-R_{1}R_{6}^{2}R_{7}-R_{1}R_{6}^{2}R_{8}-R_{2}R_{5}R_{6}^{2}-R_{2}R_{6}^{2}R_{7}-R_{2}R_{6}^{2}R_{8}+R_{5}R_{6}R_{7}^{2}$
$\displaystyle\quad+R_{6}R_{7}^{2}R_{8}+R_{7}^{2}R_{8}R_{5}-R_{7}^{2}R_{1}R_{2}-R_{1}R_{5}R_{7}^{2}-R_{1}R_{6}R_{7}^{2}-R_{1}R_{7}^{2}R_{8}-R_{2}R_{5}R_{7}^{2}$
$\displaystyle\quad-
R_{2}R_{6}R_{7}^{2}-R_{2}R_{7}^{2}R_{8}+R_{6}R_{7}R_{8}^{2}+R_{7}R_{8}^{2}R_{5}+R_{8}^{2}R_{5}R_{6}-R_{8}^{2}R_{1}R_{2}-R_{1}R_{5}R_{8}^{2}$
$\displaystyle\quad-
R_{1}R_{6}R_{8}^{2}-R_{1}R_{7}R_{8}^{2}-R_{2}R_{5}R_{8}^{2}-R_{2}R_{6}R_{8}^{2}-R_{2}R_{7}R_{8}^{2}+R_{3}R_{1}^{2}R_{2}+R_{3}R_{1}R_{2}^{2}$
$\displaystyle\quad-
R_{3}R_{5}R_{1}^{2}-R_{3}R_{5}R_{2}^{2}+R_{3}R_{6}R_{1}^{2}+R_{3}R_{6}R_{2}^{2}+R_{3}R_{7}R_{1}^{2}+R_{3}R_{7}R_{2}^{2}+R_{3}R_{8}R_{1}^{2}$
$\displaystyle\quad+R_{3}R_{8}R_{2}^{2}+R_{4}R_{1}^{2}R_{2}+R_{4}R_{1}R_{2}^{2}-R_{4}R_{5}R_{1}^{2}-R_{4}R_{5}R_{2}^{2}+R_{4}R_{6}R_{1}^{2}+R_{4}R_{6}R_{2}^{2}$
$\displaystyle\quad+R_{4}R_{7}R_{1}^{2}+R_{4}R_{7}R_{2}^{2}+R_{4}R_{8}R_{1}^{2}+R_{4}R_{8}R_{2}^{2}-3R_{1}R_{5}R_{6}R_{7}-3R_{1}R_{6}R_{7}R_{8}$
$\displaystyle\quad-3R_{1}R_{5}R_{7}R_{8}-3R_{1}R_{5}R_{6}R_{8}-3R_{2}R_{5}R_{6}R_{7}-3R_{2}R_{6}R_{7}R_{8}+3R_{5}R_{6}R_{7}R_{8}$
$\displaystyle\quad-3R_{2}R_{7}R_{8}R_{5}-3R_{2}R_{8}R_{5}R_{6}+R_{1}R_{2}R_{5}R_{6}+R_{1}R_{2}R_{5}R_{7}+R_{1}R_{2}R_{5}R_{8}$
$\displaystyle\quad-
R_{1}R_{2}R_{6}R_{7}-R_{1}R_{2}R_{6}R_{8}-R_{1}R_{2}R_{7}R_{8}-R_{3}R_{5}R_{6}R_{7}-R_{3}R_{6}R_{7}R_{8}$
$\displaystyle\quad-
R_{3}R_{7}R_{8}R_{5}-R_{3}R_{8}R_{5}R_{6}-R_{3}R_{5}R_{1}R_{2}+R_{3}R_{6}R_{1}R_{2}+R_{3}R_{7}R_{1}R_{2}$
$\displaystyle\quad+R_{3}R_{8}R_{1}R_{2}+R_{3}R_{1}R_{5}R_{6}+R_{3}R_{1}R_{5}R_{7}+R_{3}R_{1}R_{5}R_{8}+R_{3}R_{1}R_{6}R_{7}$
$\displaystyle\quad+R_{3}R_{1}R_{6}R_{8}+R_{3}R_{1}R_{7}R_{8}+R_{3}R_{2}R_{5}R_{6}+R_{3}R_{2}R_{5}R_{7}+R_{3}R_{2}R_{5}R_{8}$
$\displaystyle\quad+R_{3}R_{2}R_{6}R_{7}+R_{3}R_{2}R_{6}R_{8}+R_{3}R_{2}R_{7}R_{8}-R_{4}R_{5}R_{6}R_{7}-R_{4}R_{6}R_{7}R_{8}$
$\displaystyle\quad-
R_{4}R_{7}R_{8}R_{5}-R_{4}R_{8}R_{5}R_{6}-R_{4}R_{5}R_{1}R_{2}+R_{4}R_{6}R_{1}R_{2}+R_{4}R_{7}R_{1}R_{2}$
$\displaystyle\quad+R_{4}R_{8}R_{1}R_{2}+R_{4}R_{1}R_{5}R_{6}+R_{4}R_{1}R_{5}R_{7}+R_{4}R_{1}R_{5}R_{8}+R_{4}R_{1}R_{6}R_{7}$
$\displaystyle\quad+R_{4}R_{1}R_{6}R_{8}+R_{4}R_{1}R_{7}R_{8}+R_{4}R_{2}R_{5}R_{6}+R_{4}R_{2}R_{5}R_{7}+R_{4}R_{2}R_{5}R_{8}$
$\displaystyle\quad+R_{4}R_{2}R_{6}R_{7}+R_{4}R_{2}R_{6}R_{8}+R_{4}R_{2}R_{7}R_{8}\Big{)}.$

##  5 Concluding Remarks

In this article, we investigate the Manakov-Santini equation starting from
Lax-Sato formulation of associated hierarchy and obtain equations (25), (26),
which generalize the results of [6]. From these, one can introduce new
coordinates (34 ) such that the non-hydrodynamic evolution (32), (33) of
waterbag reduction transforms to non-homogeneous Riemann invariants form of
hydrodynamic type (40). The equation (40) is not linearly degenerate. Hence
the generalization of [9, 10] from linearly degenerate case to the general one
could be interesting. Also, the solution structures of (40) having infinite
symmetries should be investigated. These issues will be published elsewhere.

## Appendix

##  Appendix A Commutability properties of the non-homogeneous diagonal system

We start from the commutability of (41) by
$\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$:

$\displaystyle\partial_{m}\partial_{n}R^{i}$
$\displaystyle=\partial_{m}(\Lambda_{n}^{i}R^{i}_{x})+\partial_{m}Q_{n}^{i},$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\partial_{m}R^{j})R^{i}_{x}+\Lambda_{n}^{i}\partial_{x}(\partial_{m}R^{i})+\sum_{j}(\partial_{j}Q_{n}^{i})(\partial_{m}R^{j}),$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j})R^{i}_{x}+\Lambda_{n}^{i}\partial_{x}(\Lambda_{m}^{i}R^{i}_{x}+Q_{m}^{i})+\sum_{j}(\partial_{j}Q_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j}),$
$\displaystyle=\sum_{j}(\partial_{j}\Lambda_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j})R^{i}_{x}+\Lambda_{n}^{i}\Big{(}\sum_{j}(\partial_{j}\Lambda_{m}^{i})R^{j}_{x}R^{i}_{x}+\Lambda_{m}^{i}R^{i}_{xx}+\sum_{j}(\partial_{j}Q_{m}^{i})R^{j}_{x}\Big{)}$
$\displaystyle\quad+\sum_{j}(\partial_{j}Q_{n}^{i})(\Lambda_{m}^{j}R^{j}_{x}+Q_{m}^{j}),$
$\displaystyle=\sum_{j}\Big{[}(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{j}+(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{i}\Big{]}R^{j}_{x}R^{i}_{x}+\sum_{j}(\partial_{j}\Lambda_{n}^{i})Q_{m}^{j}R^{i}_{x}+\Lambda_{n}^{i}\Lambda_{m}^{i}R^{i}_{xx}$
$\displaystyle\quad+\sum_{j}\Big{[}(\partial_{j}Q_{m}^{i})\Lambda_{n}^{i}+(\partial_{j}Q_{n}^{i})\Lambda_{m}^{j}\Big{]}R^{j}_{x}+\sum_{j}(\partial_{j}Q_{n}^{i})Q_{m}^{j}.$

Similarly,

$\displaystyle\partial_{n}\partial_{m}R^{i}$ $\displaystyle=$
$\displaystyle\partial_{n}(\Lambda_{m}^{i}R^{i}_{x})+\partial_{n}Q_{m}^{i},$
$\displaystyle=$
$\displaystyle\sum_{j}\Big{[}(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{j}+(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{i}\Big{]}R^{j}_{x}R^{i}_{x}+\sum_{j}(\partial_{j}\Lambda_{m}^{i})Q_{n}^{j}R^{i}_{x}+\Lambda_{m}^{i}\Lambda_{n}^{i}R^{i}_{xx}$
$\displaystyle\quad+\sum_{j}\Big{[}(\partial_{j}Q_{n}^{i})\Lambda_{m}^{i}+(\partial_{j}Q_{m}^{i})\Lambda_{n}^{j}\Big{]}R^{j}_{x}+\sum_{j}(\partial_{j}Q_{m}^{i})Q_{n}^{j}.$

Then, $\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$ provide
the following compatibility conditions:

  * (i)

Taking the coefficients of $R^{j}_{x}R^{i}_{x}$ we have

$(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{j}+(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{i}=(\partial_{j}\Lambda_{m}^{i})\Lambda_{n}^{j}+(\partial_{j}\Lambda_{n}^{i})\Lambda_{m}^{i},$

which implies

$\frac{\partial_{j}\Lambda_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}\Lambda_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.1)

  * (ii)

Taking the coefficients of $R^{i}_{x}$ we have

$(\partial_{j}\Lambda_{n}^{i})Q_{m}^{j}=(\partial_{j}\Lambda_{m}^{i})Q_{n}^{j}.$

Combining (A.1), the above equation can be written as

$\frac{Q_{n}^{j}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{Q_{m}^{j}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.2)

  * (iii)

Taking the coefficients of $R^{j}_{x}$ we get

$\frac{\partial_{j}Q_{n}^{i}}{\Lambda_{n}^{j}-\Lambda_{n}^{i}}=\frac{\partial_{j}Q_{m}^{i}}{\Lambda_{m}^{j}-\Lambda_{m}^{i}}.$
(A.3)

  * (iv)

The zero-th term of
$\partial_{m}\partial_{n}R^{i}=\partial_{n}\partial_{m}R^{i}$ give us

$\frac{\partial_{j}Q_{n}^{i}}{Q_{n}^{j}}=\frac{\partial_{j}Q_{m}^{i}}{Q_{m}^{j}}.$
(A.4)

Notice that according to (A.2), equation (A.4) is equivalent to (A.3). To
summarize, we have three compatibility conditions (A.1), (A.2) and (A.4).

## Acknowledgments

The first author (LVB) is grateful to National Defense University (Taoyuan,
Taiwan), where a part of this work was done, for hospitality. This research
was particularly supported by the Russian-Taiwanese grant 95WFE0300007 (RFBR
grant 06-01-89507) and NSC-95-2923-M-606-001-MY3. LVB was also supported in
part by RFBR grant 07-01-00446.

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