Stationary Flows Revisited

Stationary Flows Revisited

Allan P. FORDY a and Qing HUANG b

A.P. Fordy and Q. Huang

a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
<EMAIL_ADDRESS>https://eps.leeds.ac.uk/maths/staff/4024/professor-
allan-p-fordy

b) School of Mathematics, Center for Nonlinear Studies, Northwest University,
b) Xi’an 710069, P.R. China<EMAIL_ADDRESS>
Received October 28, 2022, in final form March 08, 2023; Published online
March 29, 2023

In this paper we revisit the subject of stationary flows of Lax hierarchies of
a coupled KdV class. We explain the main ideas in the standard KdV case and
then consider the dispersive water waves (DWW) case, with respectively 2 and 3
Hamiltonian representations. Each Hamiltonian representation gives us a
different form of stationary flow. Comparing these, we construct Poisson maps,
which, being non-canonical, give rise to bi-Hamiltonian representations of the
stationary flows. An alternative approach is to use the Miura maps, which we
do in the case of the DWW hierarchy, which has two “modifications”. This
structure gives us 3 sequences of Poisson related stationary flows. We use the
Poisson maps to build a tri-Hamiltonian representation of each of the three
stationary hierarchies. One of the Hamiltonian representations allows a multi-
component squared eigenfunction expansion, which gives $N$ degrees of freedom
Hamiltonians, with first integrals. A Lax representation for each of the
stationary flows is derived from the coupled KdV matrices. In the case of 3
degrees of freedom, we give a generalisation of our Lax matrices and
Hamiltonian functions, which allows a connection with the rational
Calogero–Moser (CM) system. This gives a coupling of the CM system with other
potentials, along with a Lax representation. We present the particular case of
coupling one of the integrable Hénon–Heiles systems to CM.

KdV hierarchy; stationary flows; bi-Hamiltonian; complete integrability;
Hénon–Heiles; Calogero–Moser

35Q53; 37K10; 70H06

##  1 Introduction

It has long been known [6] that the stationary flows of nonlinear evolution
equations of “KdV-type” are themselves (finite-dimensional) completely
integrable Hamiltonian systems. Specifically, [6] considered the KdV
hierarchy, with its first Hamiltonian structure:

$\displaystyle u_{t_{n}}=\partial_{x}\delta_{u}H_{n+1},$ (1.1)

where $H_{n}$ are the KdV Hamiltonian densities. Setting $u_{t_{n}}=0$ leads
to an ODE in (generalised) Lagrangian form, with
${\mathcal{L}}_{n+1}=H_{n+1}-\alpha u$. The (generalised) Legendre
transformation then leads to canonical variable, in which the stationary flow
takes Hamiltonian form. The lower flows $u_{t_{m}}$, $m=1,\dots,n-1$, then
restrict to this finite-dimensional manifold, forming a system of commuting
Hamiltonian flows, corresponding to Hamiltonian functions which are derived
from the fluxes of the lower KdV densities.

In [3], it was shown that each of these stationary flows is bi-Hamiltonian on
an extended space, with additional dynamical variable $\alpha$. This was
derived by using the Miura map, which also gave a bi-Hamiltonian formulation
of the stationary MKdV equation. This should be compared with the full MKdV
equation, with only one local Hamiltonian structure.

In [12], the 3 known integrable cases of the Hénon–Heiles equation were shown
to be related to the stationary reductions of the 3 known fifth order (single
component) nonlinear evolution equations with a Lax representation. One of
these is in the KdV hierarchy, but not in the canonical coordinates derived
from (1.1). In this calculation the fourth order stationary flow (for one
coordinate $q_{1}$) is derived as a consequence of the coupled second order
system, by differentiation and elimination, so the relation to canonical
variables is not addressed. In [4], these canonical variables are directly
constructed by using the second Hamiltonian structure of the KdV hierarchy.
Indeed, in [4] a general approach, using squared eigenfunction coordinates,
was introduced, relating several well-known completely integrable, finite-
dimensional Hamiltonian systems to the stationary flows of various well-known
Lax hierarchies.

In this paper we revisit this subject, bringing together the ideas of [3] and
[4], enabling us to build bi- and tri-Hamiltonian representations of a number
of interesting systems. The approach also allows us to build Lax
representations for these. These systems are certainly completely integrable,
but some are superintegrable [8, 11, 16].

Since many of our formulae follow from those of coupled KdV hierarchies,
associated with “energy-dependent” Schrödinger operators [1, 2], we give a
brief overview of these in Section 2.

Our main explanation of stationary reductions is given in Section 3, in the
context of the KdV hierarchy. The two Hamiltonian representations of each PDE
flow give us two stationary manifolds with different coordinates. This can be
compared with the approach of [3], which used the Miura map and the “modified”
equation. The second of the Hamiltonian representations in this paper has a
multi-component generalisation which gives some superintegrable systems, such
as the Garnier system (3.5b) and a generalised Hénon–Heiles system (3.12b)
(both with $N$-components). Choosing $N$ so that the stationary manifolds of
the two representations have the same dimension, allows us to build Poisson
maps between these manifolds, giving us a bi-Hamiltonian representation of the
stationary flows (see Sections 3.1.1 and 3.2.1). The Lax representations of
these stationary flows are discussed in Section 3.3.

We discuss the stationary flows of the DWW hierarchy in Section 4. Following
the ideas of Section 3, we find $N$-component, superintegrable systems (4.4c)
and (4.8a), with (respectively) sextic and quartic potentials. Again, choosing
$N$ so that the stationary manifolds of the two representations have the same
dimension, allows us to build Poisson maps, giving us bi-Hamiltonian
representations (see Sections 4.1.1 and 4.2.1). Lax representations are
presented in Section 4.3.

In Section 5, we use the Miura maps of Section 2.2.1 to construct a tri-
Hamiltonian formulation of the stationary flows of the DWW hierarchy and the
two modified hierarchies, depicted in Figure 1. As this figure indicates,
there are six local Hamiltonian operators (for the PDEs), arranged in a
triangular array. In the stationary coordinates, we build a square array of
nine Poisson matrices, so each of the stationary hierarchies is tri-
Hamiltonian.

Some of the Lax representations of Sections 3.3 and 4.3 can be generalised to
incorporate an arbitrary function, which enable us to couple some of our
potentials to the rational Calogero–Moser system [7, 17]. This is discussed in
Section 6, where we specifically generalise the Garnier system (3.5b), the
generalised Hénon–Heiles system (3.12b) and the system with Hamiltonian
(4.8a). Using the approach of [13], these can all be canonically transformed
to the rational Calogero–Moser Hamiltonian, with additional potential terms.
These are all completely integrable and have a Lax representation. To
illustrate this, we explicitly present the Hénon–Heiles case.

##  2 Lax representation of coupled KdV equations

In [1, 2] a detailed analysis was given of coupled KdV equations, associated
with “energy dependent” Schrödinger operators. Here we give a very brief
review to present a few facts which we use in this paper.

The “energy dependent” Schrödinger equation is

$\displaystyle
L\psi=\big{(}\partial_{x}^{2}+u\big{)}\psi=0,\qquad\text{where}\quad
u=\sum_{i=0}^{M-1}u_{i}\lambda^{i}-\lambda^{M},$ (2.1a) such that $u_{i}$ are
functions of $x$ and some “time” parameters $t_{n}$. Suppose for one of these
(denoted “$t$”), we have
$\displaystyle\psi_{t}=A\psi=(2P\partial_{x}-P_{x})\psi.$ (2.1b) From (2.1a)
and (2.1b) we can obtain two formulae for $\psi_{xxt}$, which, when equated,
lead to $\displaystyle
u_{t}=\big{(}\partial_{x}^{3}+4u\partial_{x}+2u_{x}\big{)}P.$ (2.1c) The
coupled KdV hierarchy is given by the polynomial expansion $\displaystyle
P^{(n)}=\sum_{k=0}^{n}\lambda^{n-k}P_{k},\qquad\text{with}\quad P_{0}=1.$
(2.1d) Substituting this into (2.1c) and equating coefficients of powers of
$\lambda$ gives us a recursive formula for $P_{k}$ and then the formulae for
$u_{it_{n}}$, which can be written in matrix form to give the Hamiltonian
formulation. In this paper we only consider the KdV ($M=1$) and DWW ($M=2$)
cases, so just describe the Hamiltonian formulations for these specific cases.

This differential operator Lax pair is rewritten in “zero-curvature” form

$\displaystyle\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix}_{x}=\begin{pmatrix}0&1\\\ -u&0\\\
\end{pmatrix}\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix}=U\Psi,\qquad\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix}_{t}=\begin{pmatrix}-P_{x}&2P\\\ -2uP-P_{xx}&P_{x}\\\
\end{pmatrix}\begin{pmatrix}\psi_{1}\\\ \psi_{2}\end{pmatrix}=V\Psi,$ (2.2a)
with integrability condition $\displaystyle U_{t}-V_{x}+[U,V]=0,$ (2.2b) which
again leads to (2.1c). For a specific hierarchy, we must substitute the
correct form of $u$. For the specific $t_{n}$ flow, we must also use $P^{(n)}$
of (2.1d) in $V$, leading to $\displaystyle
U_{t_{n}}-V^{(n)}_{x}+\big{[}U,V^{(n)}\big{]}=0.$ (2.2c) We then have a direct
construction of a Lax pair for the stationary flow $U_{t_{n}}=0$, by rewriting
$V^{(n)}$ in terms of the coordinates being used on the stationary manifold.
This will be explained in more detail in Section 3.

###  2.1 The KdV hierarchy and conservation laws

The KdV hierarchy corresponds to choosing $M=1$ in (2.1a), so
$u=u_{0}-\lambda$. The coefficients $P_{k}$ in (2.1d) are constructed
recursively, the first few being

$\displaystyle P_{0}=1,\qquad P_{1}=\frac{1}{2}u_{0},\qquad
P_{2}=\frac{1}{8}\big{(}u_{0xx}+3u_{0}^{2}\big{)}.$

The flows are given by (2.1c), with $P=P^{(n)}$ and can be written in bi-
Hamiltonian form

$\displaystyle
u_{0t_{n}}=B_{1}\delta_{u_{0}}H_{n}=B_{0}\delta_{u_{0}}H_{n+1},\qquad n\geq
0,$ (2.3a) with $\displaystyle
B_{1}=\partial_{x}^{3}+4u_{0}\partial_{x}+2u_{0x},\qquad
B_{0}=4\partial_{x}\qquad\text{and}\qquad P_{n}=\delta_{u_{0}}H_{n},$ (2.3b)
where $\delta_{u_{0}}$ denotes the variational derivative with respect to
$u_{0}$.

In particular,

$\displaystyle H_{0}=u_{0},\qquad H_{1}=\frac{1}{4}u_{0}^{2},\qquad
H_{2}=\frac{1}{8}\left(u_{0}^{3}-\frac{1}{2}u_{0x}^{2}\right),$ $\displaystyle
H_{3}=\frac{1}{64}\big{(}5u_{0}^{4}-10u_{0}u_{0x}^{2}+u_{0xx}^{2}\big{)}.$
(2.3c)

Corresponding to these densities, we have an array of fluxes $F_{nm}$, in
local conservation laws, given by the $t_{n}$ evolution of $H_{m}$:

$\displaystyle\partial_{t_{n}}H_{m}=\partial_{x}F_{nm}.$ (2.3d)

In particular, we have

$\displaystyle F_{10}=\frac{1}{2}\big{(}u_{0xx}+3u_{0}^{2}\big{)},\qquad
F_{11}=\frac{1}{8}\big{(}2u_{0}u_{0xx}-u_{0x}^{2}+4u_{0}^{3}\big{)},$
$\displaystyle
F_{20}=\frac{1}{8}\big{(}10u_{0}^{3}+5u_{0x}^{2}+10u_{0}u_{0xx}+u_{0xxxx}\big{)},$
$\displaystyle
F_{21}=\frac{1}{32}\big{(}15u_{0}^{4}+20u_{0}^{2}u_{0xx}+u_{0xx}^{2}-2u_{0x}u_{0xxx}+2u_{0}u_{0xxxx}\big{)}.$
(2.3e)

###  2.2 The DWW hierarchy and conservation laws

The DWW hierarchy corresponds to choosing $M=2$ in (2.1a), so
$u=u_{0}+u_{1}\lambda-\lambda^{2}$. The coefficients $P_{k}$ in (2.1d) are
constructed recursively, the first few being

$\displaystyle P_{0}=1,\qquad P_{1}=\frac{1}{2}u_{1},\qquad
P_{2}=\frac{1}{8}\big{(}4u_{0}+3u_{1}^{2}\big{)},\qquad
P_{3}=\frac{1}{16}\big{(}12u_{0}u_{1}+5u_{1}^{3}+2u_{1xx}\big{)}.$

The general recursion, defined by (2.1c) is

$\displaystyle J_{0}P_{m-2}+J_{1}P_{m-1}+J_{2}P_{m}=0,$ (2.4a) with
$\displaystyle J_{0}=\partial_{x}^{3}+4u_{0}\partial_{x}+2u_{0x},\qquad
J_{1}=4u_{1}\partial_{x}+2u_{1x},\qquad J_{2}=-4\partial_{x}.$ For
$P=P^{(n)}$, we have $\displaystyle u_{0t_{n}}+\lambda
u_{1t_{n}}=J_{0}P_{n}+\lambda(J_{0}P_{n-1}+J_{1}P_{n}),$ (2.4b) which can be
written in vector form $\displaystyle\begin{pmatrix}u_{0}\\\ u_{1}\\\
\end{pmatrix}_{t_{n}}=\begin{pmatrix}0&J_{0}\\\ J_{0}&J_{1}\\\
\end{pmatrix}\begin{pmatrix}P_{n-1}\\\ P_{n}\\\
\end{pmatrix}=\begin{pmatrix}J_{0}&0\\\ 0&-J_{2}\\\
\end{pmatrix}\begin{pmatrix}P_{n}\\\ P_{n+1}\\\
\end{pmatrix}=\begin{pmatrix}-J_{1}&-J_{2}\\\ -J_{2}&0\\\
\end{pmatrix}\begin{pmatrix}P_{n+1}\\\ P_{n+2}\\\ \end{pmatrix},$ (2.4c)

with the first being given directly by (2.4b). The general recursion (2.4a)
then gives the second two representations. These matrix operators are the 3
compatible Hamiltonian operators of the DWW hierarchy, labelled respectively
as $B_{2}$, $B_{1}$ and $B_{0}$. Defining

$\displaystyle\begin{pmatrix}P_{n-1}\\\ P_{n}\\\
\end{pmatrix}=\delta_{u}H_{n}=\begin{pmatrix}\delta_{u_{0}}H_{n}\\\
\delta_{u_{1}}H_{n}\\\ \end{pmatrix},$ (2.5a) we have $\displaystyle{\bf
u}_{t_{n}}=B_{2}\delta_{u}H_{n}=B_{1}\delta_{u}H_{n+1}=B_{0}\delta_{u}H_{n+2},\qquad\text{for}\quad
n\geq 0,$ (2.5b) where ${\bf u}=(u_{0},u_{1})^{\rm T}$. This is the
construction given in [1, 2]. An $r$-matrix formulation was given in [20].

The above construction gives an infinite sequence of Hamiltonian functions,
$H_{n}$, the first few of which are

$\displaystyle H_{0}=u_{1},\qquad H_{1}=u_{0}+\frac{1}{4}u_{1}^{2},\qquad
H_{2}=\frac{1}{2}u_{1}\left(u_{0}+\frac{1}{4}u_{1}^{2}\right),$ $\displaystyle
H_{3}=\frac{1}{4}\left(u_{0}^{2}+\frac{3}{2}u_{0}u_{1}^{2}+\frac{5}{16}u_{1}^{4}-\frac{1}{4}u_{1x}^{2}\right).$
(2.5c)

We have

$\displaystyle
B_{0}\delta_{u}H_{0}=B_{0}\delta_{u}H_{1}=B_{1}\delta_{u}H_{0}=0,$ (2.5d)

meaning that $B_{0}$ has two local Casimir functions $H_{0}$, $H_{1}$, whilst
$B_{1}$ has one local Casimir function $H_{0}$.

We define fluxes $F_{nm}$ by the same formula (2.3d), giving

$\displaystyle F_{10}=2u_{0}+\frac{3}{2}u_{1}^{2},\qquad
F_{11}=\frac{1}{2}\big{(}4u_{0}u_{1}+u_{1}^{3}+u_{1xx}\big{)},$ $\displaystyle
F_{12}=\frac{1}{32}\big{(}16u_{0}^{2}+40u_{0}u_{1}^{2}+9u_{1}^{4}-4u_{1x}^{2}+8u_{1}u_{1xx}\big{)},\qquad
F_{20}=3u_{0}u_{1}+\frac{5}{4}u_{1}^{3}+\frac{1}{2}u_{1xx},$ $\displaystyle
F_{21}=\frac{1}{32}\big{(}48u_{0}^{2}+72u_{0}u_{1}^{2}+15u_{1}^{4}+20u_{1x}^{2}+16u_{0xx}+32u_{1}u_{1xx}\big{)},$
$\displaystyle
F_{22}=\frac{1}{32}\big{(}48u_{0}^{2}u_{1}+9u_{1}^{5}-8u_{0x}u_{1x}+8u_{1}u_{0xx}+18u_{1}^{2}u_{1xx}+8u_{0}\big{(}6u_{1}^{3}+u_{1xx}\big{)}\big{)}.$
(2.5e)

####  2.2.1 Miura maps

In [2], Miura maps were presented for the entire class of systems described by
the Lax operator (2.1a). In the DWW case, there are 3 sets of variables
$(u_{0},u_{1})$, $(w_{0},w_{1})$ and $(v_{0},v_{1})$, related by

$\displaystyle u_{0}=-w_{0x}-w_{0}^{2},\qquad
u_{1}=w_{1}\qquad\text{and}\qquad w_{0}=v_{0},\qquad
w_{1}=-v_{1x}-2v_{0}v_{1}$ $\displaystyle\qquad{}\Rightarrow\quad
u_{0}=-v_{0x}-v_{0}^{2},\qquad u_{1}=-v_{1x}-2v_{0}v_{1}.$ (2.6)

In the $u$-space we have the 3 local Hamiltonian operators $B_{i}^{u}\equiv
B_{i}$, $i=0,1,2$ given by (2.4c), with 2 local operators $B_{2}^{w}$,
$B_{1}^{w}$ in the $w$-space and just 1 local operator $B_{2}^{v}$ in the
$v$-space. These are depicted in Figure 1 and related by

$\displaystyle
B_{k}^{u}=\frac{Du}{Dw}B_{k}^{w}\left(\frac{Du}{Dw}\right)^{\dagger},\qquad\text{for}\quad
k=1,2,\qquad\text{and}\qquad
B_{2}^{w}=\frac{Dw}{Dv}B_{2}^{v}\left(\frac{Dw}{Dv}\right)^{\dagger},$

where $\frac{Du}{Dw}$ and $\frac{Dw}{Dv}$ are the Jacobians of the maps (2.6),
with

$\displaystyle
B_{2}^{w}=\begin{pmatrix}0&-\partial_{x}(\partial_{x}-2w_{0})\\\
(\partial_{x}+2w_{0})\partial_{x}&4w_{1}\partial_{x}+2w_{1x}\\\
\end{pmatrix},$ $\displaystyle B_{2}^{v}=\begin{pmatrix}0&-\partial_{x}\\\
-\partial_{x}&0\\\ \end{pmatrix},\qquad
B_{1}^{w}=\begin{pmatrix}-\partial_{x}&0\\\ 0&4\partial_{x}\\\ \end{pmatrix}.$

$B_{2}^{u}$$B_{1}^{u}$$B_{0}^{u}$$B_{2}^{w}$$B_{1}^{w}$$B_{2}^{v}$ Figure 1:
Hamiltonian operators in the 3 spaces.

We have built a sequence of Hamiltonians $H_{k}^{u}\equiv
H_{k}^{u}[u_{0},u_{1}]$, as functions of $u_{0}$, $u_{1}$ and their
$x$-derivatives. The Miura maps then define $H_{k}^{w}$ and $H_{k}^{v}$, by
substituting the formulae (2.6) into $H_{k}^{u}$. The flow ${\bf u}_{t_{n}}$,
defined by (2.5b), gives rise to the flows:

$\displaystyle{\bf
w}_{t_{n}}=B_{2}^{w}\delta_{w}H_{n}^{w}=B_{1}^{w}\delta_{w}H_{n+1}^{w}\qquad\text{and}\qquad{\bf
v}_{t_{n}}=B_{2}^{v}\delta_{v}H_{n}^{v},$ (2.7)

where ${\bf w}=(w_{0},w_{1})^{\rm T}$ and ${\bf v}=(v_{0},v_{1})^{\rm T}$.

##  3 KdV stationary flows

In this section we discuss the two different canonical representations of the
stationary flows, related to the two Hamiltonian representations of the KdV
hierarchy. This gives us a new way of constructing a bi-Hamiltonian
representation of the stationary flows. We also discuss the Lax formulation of
the derived systems.

A stationary flow (for $t_{n}$) means that we reduce to a finite-dimensional
space with $u_{0t_{n}}=0$. The “time” variable for this system is $x$, which
is the variable which appears in the Lagrangians, given below.

This gives an ODE, defined by one of the two representations given in (2.3a):

  1. 1.

Using $B_{0}$ with the Bogoyavlensky–Novikov coordinates [6], we build the
first Lagrangian:

$\displaystyle
B_{0}\delta_{u_{0}}H_{n+1}=0\quad\Rightarrow\quad\delta_{u_{0}}(H_{n+1}-\alpha
u_{0})=0\quad\Rightarrow\quad{\mathcal{L}}_{n+1}=H_{n+1}-\alpha u_{0}.$ (3.1)

We then use the (generalised) Legendre transformation to find canonical
coordinates $(q_{i},p_{i})$ and the Hamiltonian function.

  2. 2.

Using $B_{1}$ and the squared eigenfunction representation (following [4]):

$\displaystyle\delta_{u_{0}}H_{n}=a\varphi^{2},$ (3.2a) and since $B_{1}$ is
skew symmetric, we can show
$\displaystyle\varphi^{2}\big{(}B_{1}\varphi^{2}\big{)}=\left(2\varphi^{3}(\varphi_{xx}+u_{0}\varphi)\right)_{x}=0\quad\Rightarrow\quad
2\varphi^{3}(\varphi_{xx}+u_{0}\varphi)=4\beta.$ (3.2b) Equations (3.2a) and
(3.2b) can then be written as variational derivatives of a single Lagrangian
function
$\displaystyle{\mathcal{L}}_{n}=\frac{1}{2a}H_{n}-\frac{1}{2}u_{0}\varphi^{2}+\frac{1}{2}\varphi_{x}^{2}-\frac{\beta}{\varphi^{2}}.$
(3.2c)

Again, the (generalised) Legendre transformation gives canonical coordinates
$(Q_{i},P_{i})$ and the Hamiltonian function.

######  Remark 3.1.

Clearly, the algorithm for constructing the $H_{n}$ of (2.3c) leads to awkward
looking overall factors. In the above formulae for ${\mathcal{L}}_{n}$ and
${\mathcal{L}}_{n+1}$, we always choose convenient multiples of $H_{n}$ and
$H_{n+1}$.

######  Remark 3.2.

In equation (3.2a), the parameter $a$ is arbitrary, but we can always rescale
$\varphi$, so choose ${a=\frac{1}{2}}$.

Each of the above representations gives an $n$-degrees of freedom canonical
Hamiltonian system for the stationary flow $u_{0t_{n}}=0$. We can extend the
spaces to $2n+1$ dimensions, by adding the arbitrary constants $\alpha$ and
$\beta$ as dynamical variables, which are just Casimir functions of the
respective extended canonical brackets.

Poisson map. We can then write $q_{i}$, $p_{i}$, $\alpha$ in terms of
$u_{0}(x)$ and its derivatives, which in turn can be written in terms of
$Q_{i}$, $P_{i}$, $\beta$ and vice versa, thus giving us a Poisson map between
the two systems. We emphasise that this is _not_ a canonical transformation,
so generates a second Poisson bracket for each of the systems. We give a more
detailed description of the procedure for the simplest case of the $t_{1}$
flow.

First integrals. Setting $u_{0t_{n}}=0$ in (2.3d) means that the fluxes
$F_{nm}$ are first integrals. On this (extended) stationary manifold, the
$n+1$ functions $\\{F_{nk}\\}_{k=0}^{n}$ are independent.

Multi-component squared eigenfunctions. Since $B_{1}$ is a linear operator, we
can extend the squared eigenfunction representation (3.2a) to include multiple
eigenfunctions, with

$\displaystyle\delta_{u_{0}}H_{n}=\frac{1}{2}\sum_{i}\varphi_{i}^{2}.$ (3.3a)

The first option is to repeat the calculation (3.2a) for each $\varphi_{i}$,
to obtain $2\varphi_{i}^{3}(\varphi_{ixx}+u_{0}\varphi_{i})=4\beta_{i}$,
giving

$\displaystyle{\mathcal{L}}_{n}=H_{n}+\sum_{i}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{\beta_{i}}{\varphi_{i}^{2}}\right).$
(3.3b)

For the second option we first define $\varphi^{2}=\sum_{i}\varphi_{i}^{2}$.
We then consider

$\displaystyle\varphi^{2}B_{1}\varphi^{2}=0\quad\Rightarrow\quad\varphi^{2}\sum_{i}\varphi_{i}(\varphi_{ixx}+u_{0}\varphi_{i})+\sum_{i<j}(\varphi_{i}\varphi_{jx}-\varphi_{j}\varphi_{ix})^{2}=\text{const}.$
(3.3c)

If we now set

$\displaystyle\varphi_{ixx}+u_{0}\varphi_{i}=\frac{2\beta\varphi_{i}}{\varphi^{4}},\qquad\text{then}\quad\varphi^{2}\sum_{i}\varphi_{i}(\varphi_{ixx}+u_{0}\varphi_{i})=2\beta.$
(3.3d)

Furthermore, $(\varphi_{i}\varphi_{jx}-\varphi_{j}\varphi_{ix})_{x}=0$, for
each $i<j$, so the expression in (3.3c) really is a constant. Therefore,
(3.3d) is a solution to our problem. We then have

$\displaystyle{\mathcal{L}}_{n}=H_{n}+\frac{1}{2}\sum_{i}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi^{2}-\frac{\beta}{\varphi^{2}}.$
(3.3e)

It can be seen that this Lagrangian is rotationally invariant in the $\varphi$
space and that the expressions
$(\varphi_{i}\varphi_{jx}-\varphi_{j}\varphi_{ix})$ are just the angular
momenta, which are constants of the motion.

###  3.1 The $\boldsymbol{t_{1}}$ flow

We build 3 different Hamiltonians.

The Lagrangian (3.1), using $8H_{2}$ and labelling $-{\mathcal{L}}_{2}$ as
$\mathcal{L}$, gives

$\displaystyle{\mathcal{L}}=\frac{1}{2}u_{0x}^{2}-u_{0}^{3}+\alpha
u_{0}\qquad\Rightarrow\quad h^{(q)}=\frac{1}{2}p_{1}^{2}+q_{1}^{3}-\alpha
q_{1}.$ (3.4)

The Lagrangians (3.3b) and (3.3e) are degenerate, since $H_{1}$ is independent
of $u_{0x}$. Choosing $-H_{1}$ in (3.3b) gives

$\displaystyle\tilde{\mathcal{L}}=\frac{1}{2}\sum_{i=1}^{N}\left(\varphi_{ix}^{2}-u_{0}\varphi_{i}^{2}-\frac{2\beta_{i}}{\varphi_{i}^{2}}\right)-\frac{1}{4}u_{0}^{2}.$
(3.5a) We find
$\delta_{u_{0}}\tilde{\mathcal{L}}=-\frac{1}{2}\big{(}\sum_{i=1}^{N}\varphi_{i}^{2}+u_{0}\big{)}=0$,
so substitute $u_{0}=-\sum_{i=1}^{N}\varphi_{i}^{2}$, to obtain the Garnier
system (see [22])
$\displaystyle\tilde{\mathcal{L}}=\frac{1}{2}\sum_{i=1}^{N}\varphi_{x}^{2}+\frac{1}{4}\left(\sum_{i=1}^{N}\varphi_{i}^{2}\right)^{2}-\sum_{i=1}^{N}\frac{\beta_{i}}{\varphi_{i}^{2}}$
$\displaystyle\qquad{}\Rightarrow\quad
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)-\frac{1}{4}\left(\sum_{i=1}^{N}Q_{i}^{2}\right)^{2}.$
(3.5b)

A similar calculation for the case (3.3e) leads to

$\displaystyle
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}-\frac{1}{4}\left(\sum_{i=1}^{N}Q_{i}^{2}\right)^{2}+\frac{\beta}{{\bf
Q}^{2}},$ (3.5c)

where ${\bf Q}^{2}=\sum_{i=1}^{N}Q_{i}^{2}$.

First integrals for the Hamiltonian $\boldsymbol{h^{(Q)}}$ of (3.5b). The
fluxes (2.3e) give us two integrals for this Hamiltonian, one of which is
$h^{(Q)}$ itself. The other is a deformation of the rotational Casimir in $N$
dimensions. To describe this, we define

$\displaystyle
h_{ij}=(Q_{i}P_{j}-Q_{j}P_{i})^{2}+2\left(\frac{\beta_{i}Q_{j}^{2}}{Q_{i}^{2}}+\frac{\beta_{j}Q_{i}^{2}}{Q_{j}^{2}}\right)\qquad\text{for}\quad
1\leq i<j\leq N.$ (3.6a) We then have $\displaystyle F_{10}=-2h^{(Q)},\qquad
F_{11}=\Phi^{(Q)}=\frac{1}{2}\sum_{i<j}h_{ij}+\sum_{i=1}^{N}\beta_{i}.$ (3.6b)
We see from (3.6b) that $h_{12}$ is an integral when $N=2$. In fact, for
arbitrary $N$, each $h_{ij}$ is a first integral, for all $i<j$, so for $N\geq
3$, this system is superintegrable, as is known [5, 22].

The Poisson relations for $h_{ij}$ are particularly simple when $N=3$. We
first define a cubic element $h_{123}$:

$\displaystyle
h_{123}=(Q_{1}P_{2}-Q_{2}P_{1})(Q_{2}P_{3}-Q_{3}P_{2})(Q_{3}P_{1}-Q_{1}P_{3})$
$\displaystyle\hphantom{h_{123}=}{}-2\left(\\!\frac{\beta_{1}Q_{2}Q_{3}(Q_{2}P_{3}\\!-Q_{3}P_{2})}{Q_{1}^{2}}+\frac{\beta_{2}Q_{1}Q_{3}(Q_{3}P_{1}\\!-Q_{1}P_{3})}{Q_{2}^{2}}+\frac{\beta_{3}Q_{1}Q_{2}(Q_{1}P_{2}\\!-Q_{2}P_{1})}{Q_{3}^{2}}\\!\right)\\!.$

We then have

$\displaystyle\\{h_{12},h_{23}\\}=-\\{h_{12},h_{13}\\}=-\\{h_{13},h_{23}\\}=4h_{123},$

and

$\displaystyle\\{h_{12},h_{123}\\}=2\left(h_{12}(h_{13}-h_{23})-4\beta_{1}h_{23}+4\beta_{2}h_{13}\right),$
$\displaystyle\\{h_{23},h_{123}\\}=2\left(h_{23}(h_{12}-h_{13})-4\beta_{2}h_{13}+4\beta_{3}h_{12}\right),$
$\displaystyle\\{h_{13},h_{123}\\}=2\left(h_{13}(h_{23}-h_{12})-4\beta_{3}h_{12}+4\beta_{1}h_{23}\right).$

######  Remark 3.3 (rotationally symmetric case).

The Hamiltonian $h^{(Q)}$ of (3.5c) is clearly rotationally invariant, so the
integrals (3.6a) are replaced by the angular momenta and $\Phi^{(Q)}$ just
gives the rotational Casimir in $N$ dimensions.

####  3.1.1 Bi-Hamiltonian formulation when $\boldsymbol{N=1}$

Here we have

$\displaystyle
h^{(Q)}=\frac{1}{2}P_{1}^{2}-\frac{1}{4}Q_{1}^{4}+\frac{\beta}{Q_{1}^{2}}.$
(3.7)

The Hamiltonian formulations (3.4) and (3.7) give coordinates ${\bf
q}\\!=\\!(q_{1},p_{1},\alpha)$ and ${\bf Q}\\!=\\!(Q_{1},P_{1},\beta)$,
respectively. From the definitions of the canonical coordinates, we have

$\displaystyle q_{1}=u_{0}=-Q_{1}^{2},\qquad
p_{1}=u_{0x}=-2Q_{1}P_{1},\qquad\alpha=u_{0xx}+3u_{0}^{2}=-4h^{(Q)}.$ (3.8)

In the 3D space with coordinates $\bf q$, we introduce the degenerate
extension of the canonical Poisson tensor:

$\displaystyle{\mathcal{P}}_{0}^{(q)}=\left(\begin{matrix}\phantom{-}0&1&0\\\
-1&0&0\\\ \phantom{-}0&0&0\end{matrix}\right),\qquad\text{with}\quad\frac{{\rm
d}{\bf q}}{{\rm d}t_{f}}={\mathcal{P}}_{0}^{(q)}\nabla_{q}f,$

for any function $f({\bf q})$. The formulae (3.8) represent a mapping from the
$\bf Q$ space to the $\bf q$ space, with Jacobian $\frac{\partial{\bf
q}}{\partial{\bf Q}}$. We use the inverse map to construct a Poisson tensor
${\mathcal{P}}_{0}^{(Q)}$ in the $\bf Q$ space. The Jacobian of this inverse
is just the inverse matrix of $\frac{\partial{\bf q}}{\partial{\bf Q}}$, which
is conveniently written in terms of the $\bf Q$ coordinates:

$\displaystyle{\mathcal{P}}_{0}^{(Q)}=\left(\frac{\partial{\bf
q}}{\partial{\bf
Q}}\right)^{-1}{\mathcal{P}}_{0}^{(q)}\left(\left(\frac{\partial{\bf
q}}{\partial{\bf Q}}\right)^{-1}\right)^{\rm
T},\qquad\text{with}\quad\frac{{\rm d}{\bf Q}}{{\rm
d}t_{f}}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\tilde{f},$

where $\tilde{f}({\bf Q})=f({\bf q}({\bf Q}))$. From (3.8), we find (up to
numerical factor)

$\displaystyle{\mathcal{P}}_{0}^{(Q)}=\left(\begin{matrix}\phantom{-}0&\frac{1}{Q_{1}^{2}}&-P_{1}\\\
-\frac{1}{Q_{1}^{2}}&0&-\frac{2\beta+Q_{1}^{6}}{Q_{1}^{3}}\\\
\phantom{-}P_{1}&\frac{2\beta+Q_{1}^{6}}{Q_{1}^{3}}&\phantom{-}0\end{matrix}\right),\qquad\text{with}\quad\alpha=-4h^{(Q)},\quad
h^{(q)}=-4\beta.$

This is compatible with the canonical bracket on the ${\bf Q}$ space:

$\displaystyle{\mathcal{P}}_{1}^{(Q)}=\left(\begin{matrix}\phantom{-}0&1&0\\\
-1&0&0\\\ \phantom{-}0&0&0\end{matrix}\right),$

which in turn allows us to construct a Poisson tensor
${\mathcal{P}}_{1}^{(q)}$, using $\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}{\mathcal{P}}_{1}^{(Q)}\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}^{\rm T}=-4{\mathcal{P}}_{1}^{(q)}$, which should be written in
terms of the $\bf q$ variables:

$\displaystyle{\mathcal{P}}_{1}^{(q)}=\left(\begin{matrix}0&q_{1}&p_{1}\\\
-q_{1}&0&\alpha-3q_{1}^{2}\\\ -p_{1}&3q_{1}^{2}-\alpha&0\end{matrix}\right).$

We use the two independent fluxes from (2.3e):

$\displaystyle F_{10}=\frac{1}{2}\alpha=-2h^{(Q)},\qquad
F_{11}=-\frac{1}{4}h^{(q)}=\beta,$ to build the bi-Hamiltonian ladders, which
satisfy $\displaystyle{\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha=0,\qquad{\bf
q}_{t_{h}}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},\qquad{\mathcal{P}}_{1}^{(q)}\nabla_{q}h^{(q)}=0,$
$\displaystyle{\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0,\qquad{\bf
Q}_{t_{h}}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}(-\beta),\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta=0.$
Whereas $\alpha$ and $\beta$ were chosen to be the Casimirs of the canonical
brackets, we see that $h^{(q)}$ and $h^{(Q)}$ are the Casimirs of the “second”
brackets.

###  3.2 The $\boldsymbol{t_{2}}$ flow

We build 3 different Hamiltonians.

The Lagrangian (3.1), with $32H_{3}$ and labelling ${\mathcal{L}}_{3}$ as
$\mathcal{L}$, gives

$\displaystyle{\mathcal{L}}=\frac{1}{2}\left(5u_{0}^{4}-10u_{0}u_{0x}^{2}+u_{0xx}^{2}\right)-\alpha
u_{0}$ $\displaystyle\qquad\Rightarrow\quad
h^{(q)}=\frac{1}{2}p_{2}^{2}+q_{2}p_{1}+5q_{1}q_{2}^{2}-\frac{5}{2}q_{1}^{4}+\alpha
q_{1},$ (3.10)

where we have used the generalised Legendre transformation

$\displaystyle q_{1}=u_{0},\qquad q_{2}=u_{0x},\qquad
p_{1}=\frac{\partial{\mathcal{L}}}{\partial u_{0x}}-\frac{{\rm d}}{{\rm
d}x}\left(\frac{\partial{\mathcal{L}}}{\partial
u_{0xx}}\right)=-10u_{0}u_{0x}-u_{0xxx},$ $\displaystyle
p_{2}=\frac{\partial{\mathcal{L}}}{\partial u_{0xx}}=u_{0xx}.$ (3.11)

The Lagrangian (3.3b), with $-8H_{2}$, gives

$\displaystyle\tilde{\mathcal{L}}=\frac{1}{2}\left(u_{0x}^{2}+\sum_{i=1}^{N-1}\varphi_{ix}^{2}\right)-u_{0}^{3}-\frac{1}{2}u_{0}\sum_{i=1}^{N-1}\varphi_{i}^{2}-\sum_{i=1}^{N-1}\frac{\beta_{i+1}}{\varphi_{i}^{2}}.$
(3.12a) Defining $Q_{1}=u_{0}$, $P_{1}=u_{0x}$, $Q_{i+1}=\varphi_{i}$,
$P_{i+1}=\varphi_{ix}$, $i=1,\dots,N-1$, we obtain a generalised Hénon–Heiles
system $\displaystyle
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}+\frac{1}{2}Q_{1}\left(2Q_{1}^{2}+\sum_{i=2}^{N}Q_{i}^{2}\right)+\sum_{i=2}^{N}\frac{\beta_{i}}{Q_{i}^{2}}.$
(3.12b) The fluxes (2.3e) then give three integrals $\displaystyle
4F_{20}=-h^{(Q)},\qquad
16F_{21}=f^{(Q)}=P_{1}\sum_{i=1}^{N}Q_{i}P_{i}-2Q_{1}h^{(Q)}+\frac{1}{8}\big{(}16Q_{1}^{4}+12Q_{1}^{2}{\bf
Q}^{2}+{\bf Q}^{4}\big{)},$ $\displaystyle
64F_{22}=\Phi^{(Q)}=\frac{1}{2}\sum_{2\leq i<j\leq
N}h_{ij}+\sum_{i=2}^{N}\beta_{i},$ (3.12c) where $h_{ij}$ are given by (3.6a)
and ${\bf Q}^{2}=\sum_{i=2}^{N}Q_{i}^{2}$. Again, each $h_{ij}$ is a first
integral.

For the case (3.3e), an identical calculation leads to

$\displaystyle
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}+\frac{1}{2}Q_{1}\left(2Q_{1}^{2}+\sum_{i=2}^{N}Q_{i}^{2}\right)+\frac{\beta}{{\bf
Q}^{2}}.$

This Hamiltonian is clearly rotationally invariant (in $Q_{2},\dots,Q_{N}$
space), so the integrals (3.6a) are replaced by the angular momenta and
$F_{22}$ just gives the rotational Casimir in $(N-1)$ dimensions.

####  3.2.1 Bi-Hamiltonian formulation when $\boldsymbol{N=2}$

Here we have the usual 2 degrees of freedom Hénon–Heiles Hamiltonian

$\displaystyle
h^{(Q)}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}\big{)}+\frac{1}{2}Q_{1}\big{(}2Q_{1}^{2}+Q_{2}^{2}\big{)}+\frac{\beta}{Q_{2}^{2}}.$
(3.13)

The Hamiltonian formulations (3.10) and (3.13) gave coordinates ${\bf
q}=(q_{i},p_{i},\alpha)$ and ${\bf Q}=(Q_{i},P_{i},\beta)$, respectively. From
the definitions of the canonical coordinates, we have (3.11) and
$\alpha=10u_{0}^{3}+5u_{0x}^{2}+10u_{0}u_{0xx}+u_{0xxxx}$. Since
$u_{0}=Q_{1}$, we can use $h^{(Q)}$ to calculate all $x$-derivatives:
$u_{0}^{(m+1)}=\big{\\{}u_{0}^{(m)},h^{(Q)}\big{\\}}$, where $u_{0}^{(i)}$ is
the $i^{\rm th}$ derivative of $u_{0}$ with respect to $x$, giving

$\displaystyle q_{1}=Q_{1},\qquad q_{2}=P_{1},\qquad
p_{1}=-4Q_{1}P_{1}+Q_{2}P_{2},\qquad p_{2}=-3Q_{1}^{2}-\frac{1}{2}Q_{2}^{2},$
$\displaystyle\alpha=-2h^{(Q)}.$ (3.14)

The three independent fluxes from (2.3e) give

$\displaystyle 4F_{20}=\frac{1}{2}\alpha=-h^{(Q)},\qquad
16F_{21}=h^{(q)}=f^{(Q)},\qquad 64F_{22}=-\frac{1}{2}f^{(q)}=\beta,$ (3.15a)
where $\displaystyle
f^{(q)}=p_{1}^{2}+4q_{1}p_{2}^{2}+8q_{1}q_{2}p_{1}+2\big{(}10q_{1}^{3}-q_{2}^{2}\big{)}p_{2}+2q_{1}^{2}\big{(}12q_{1}^{3}+5q_{2}^{2}\big{)}-2\alpha\big{(}p_{2}+3q_{1}^{2}\big{)},$
(3.15b) $\displaystyle
f^{(Q)}=P_{2}(Q_{2}P_{1}-Q_{1}P_{2})+\frac{1}{8}Q_{2}^{2}\big{(}4Q_{1}^{2}+Q_{2}^{2}\big{)}-\frac{2\beta
Q_{1}}{Q_{2}^{2}}.$ (3.15c)

In the 5D space with coordinates $\bf q$, we introduce a degenerate extension
of the canonical Poisson tensor and use (3.14) as a Poisson map to obtain
${\mathcal{P}}_{0}^{(Q)}$:

$\displaystyle{\mathcal{P}}_{0}^{(q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&1&0&0\\\
\phantom{-}0&\phantom{-}0&0&1&0\\\ -1&\phantom{-}0&0&0&0\\\
\phantom{-}0&-1&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&0&0\end{matrix}\right)\quad\Rightarrow\quad{\mathcal{P}}_{0}^{(Q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{Q_{2}}&a_{15}\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{Q_{2}}&-\frac{2Q_{1}}{Q_{2}^{2}}&a_{25}\\\\[5.69054pt]
\phantom{-}0&-\frac{1}{Q_{2}}&\phantom{-}0&\phantom{-}\frac{P_{2}}{Q_{2}^{2}}&a_{35}\\\\[5.69054pt]
-\frac{1}{Q_{2}}&\frac{2Q_{1}}{Q_{2}^{2}}&-\frac{P_{2}}{Q_{2}^{2}}&\phantom{-}0&a_{45}\\\\[5.69054pt]
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0\end{matrix}\right),$

where the column $(a_{15},a_{25},a_{35},a_{45},0)^{\rm
T}=-P_{1}^{(Q)}\nabla_{Q}f^{(Q)}$, with ${\mathcal{P}}_{1}^{(Q)}$ being the
compatible canonical bracket on the $\bf Q$ space, which is similarly
transformed to the $\bf q$ space:

$\displaystyle{\mathcal{P}}_{1}^{(Q)}=\begin{pmatrix}\phantom{-}0&\phantom{-}0&1&0&0\\\
\phantom{-}0&\phantom{-}0&0&1&0\\\ -1&\phantom{-}0&0&0&0\\\
\phantom{-}0&-1&0&0&0\\\ \phantom{-}0&\phantom{-}0&0&0&0\end{pmatrix}$
$\displaystyle\qquad\Rightarrow\quad{\mathcal{P}}_{1}^{(q)}=\begin{pmatrix}\phantom{-}0&1&-4q_{1}&\phantom{-}0&b_{15}\\\
-1&0&\phantom{-}4q_{2}&\phantom{-}6q_{1}&b_{25}\\\
\phantom{-}4q_{1}&-4q_{2}&\phantom{-}0&-30q_{1}^{2}-2p_{2}&b_{35}\\\
\phantom{-}0&-6q_{1}&30q_{1}^{2}+2p_{2}&\phantom{-}0&b_{45}\\\
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0\end{pmatrix},$

where the column $(b_{15},b_{25},b_{35},b_{45},0)^{\rm
T}=-2P_{0}^{(q)}\nabla_{q}h^{(q)}$.

The Hamiltonian ladders are

$\displaystyle{\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha=0,\qquad{\mathcal{P}}_{1}^{(q)}\nabla_{q}\bigl{(}-\tfrac{\alpha}{2}\bigr{)}={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},$
$\displaystyle{\mathcal{P}}_{1}^{(q)}\nabla_{q}h^{(q)}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\big{(}\tfrac{1}{2}f^{(q)}\big{)},\qquad{\mathcal{P}}_{1}^{(q)}\nabla_{q}f^{(q)}=0,$
$\displaystyle{\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0,\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}f^{(Q)},$
$\displaystyle{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}(-\beta),\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta=0.$

The two nontrivial flows correspond to the two lower commuting flows of the
KdV hierarchy. We see this by looking at the $u_{0}$ ($=q_{1}=Q_{1}$)
component of the flows.

In the $\bf q$ space, the flows ${\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}$ and
${\mathcal{P}}_{0}^{(q)}\nabla_{q}\bigl{(}-\frac{1}{2}f^{(q)}\bigr{)}$ give

$\displaystyle q_{1t_{h}}=\frac{\partial h^{(q)}}{\partial
p_{1}}=q_{2}=q_{1x},\qquad q_{1t_{f}}=\frac{\partial}{\partial
p_{1}}\bigl{(}-\tfrac{1}{2}f^{(q)}\bigr{)}=-p_{1}-4q_{1}q_{2}=q_{1xxx}+6q_{1}q_{1x}.\\!\\!\\!$
(3.16a) In the $\bf Q$ space, the flows
${\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}$ and
${\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\left(-f^{(Q)}\right)$ give $\displaystyle
Q_{1t_{h}}=\frac{\partial h^{(Q)}}{\partial P_{1}}=P_{1}=Q_{1x},\qquad
Q_{1t_{f}}=-\frac{\partial f^{(Q)}}{\partial
P_{1}}=-Q_{2}P_{2}=\big{(}Q_{1xx}+3Q_{1}^{2}\big{)}_{x}.$ (3.16b)

###  3.3 The Lax representation

As we previously said, to obtain a Lax representation for stationary flows, we
use the zero curvature representation, given in (2.2a) and (2.2c), with
$U_{t_{n}}=0$. When written in terms of the coordinates on the stationary
manifold, the characteristic equation, $\det\big{(}zI-L^{(n)}\big{)}=0$, is
$\lambda$-dependent, giving the same integrals as derived from the fluxes
$F_{ij}$.

####  3.3.1 The $\boldsymbol{t_{1}}$ flow

In this case our matrices are

$\displaystyle U=\left(\begin{matrix}0&1\\\ \lambda-
u_{0}&0\end{matrix}\right),\qquad
V^{(1)}=\frac{1}{2}\left(\begin{matrix}-u_{0x}&4\lambda+2u_{0}\\\
4\lambda^{2}-2\lambda u_{0}-2u_{0}^{2}-u_{0xx}&u_{0x}\end{matrix}\right).$
(3.17)

Using the coordinates of (3.4), we define

$\displaystyle L^{(1)}=-2V^{(1)}=\begin{pmatrix}p_{1}&-2(2\lambda+q_{1})\\\
\alpha-q_{1}^{2}+2\lambda q_{1}-4\lambda^{2}&-p_{1}\\\ \end{pmatrix}$
$\displaystyle\qquad\Rightarrow\quad
z^{2}-2h^{(q)}+4\alpha\lambda-16\lambda^{3}=0,$

where $\alpha$ and $h^{(q)}$ are constants of the motion. The Lax
representation of the equation generated by $h^{(q)}$ is

$\displaystyle
L^{(1)}_{x}=\big{\\{}L^{(1)},h^{(q)}\big{\\}}=\big{[}U,L^{(1)}\big{]},$

using the canonical Poisson bracket, acting componentwise on the matrix.

Using the coordinates of (3.5b), we have

$\displaystyle
L^{(1)}=V^{(1)}=\begin{pmatrix}\displaystyle\sum_{1}^{N}Q_{i}P_{i}&2\lambda-{\bf
Q}^{2}\\\
\displaystyle\sum_{1}^{N}\\!\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)\\!+\lambda\big{(}2\lambda+{\bf
Q}^{2}\big{)}&-\displaystyle\sum_{1}^{N}Q_{i}P_{i}\\\ \end{pmatrix}$
$\displaystyle\qquad\Rightarrow\quad z^{2}-4\lambda
h^{(Q)}\\!+2\Phi^{(Q)}\\!-4\lambda^{3}=0,$ (3.18a) where ${\bf
Q}^{2}=\sum_{i=1}^{N}Q_{i}^{2}$, with $h^{(Q)}$ and $\Phi^{(Q)}$ as given in
(3.5b) and (3.6b). The Lax equations generated by $h^{(Q)}$ and $\Phi^{(Q)}$
are $\displaystyle
L^{(1)}_{x}=\big{\\{}L^{(1)},h^{(Q)}\big{\\}}=\big{[}U,L^{(1)}\big{]},\qquad\big{\\{}L^{(1)},\Phi^{(Q)}\big{\\}}=0.$
(3.18b)

In fact, $\big{\\{}L^{(1)},h_{ij}\big{\\}}=0$, for each of the functions
$h_{ij}$ of (3.6a).

When $N=1$, the function $\Phi^{(Q)}$ is replaced by $\beta$.

######  Remark 3.4 (rotationally symmetric case).

Similar equations hold for the case (3.5c), with
$\sum_{1}^{N}\frac{\beta_{i}}{Q_{i}^{2}}$ replaced by $\frac{\beta}{{\bf
Q}^{2}}$ and the integral $\Phi^{(Q)}$ by the rotational Casimir.

####  3.3.2 The $\boldsymbol{t_{2}}$ flow

In this case our principal matrix is proportional to $V^{(2)}$, which is to be
written in terms of appropriate coordinates.

Using the coordinates from (3.10), we define

$\displaystyle L^{(2)}=-8V^{(2)}=\left(\begin{matrix}4\lambda
q_{2}-p_{1}-4q_{1}q_{2}&-2\big{(}8\lambda^{2}+4\lambda
q_{1}+p_{2}+3q_{1}^{2}\big{)}\\\ a_{21}&p_{1}+4q_{1}q_{2}-4\lambda
q_{2}\end{matrix}\right),$

where
$a_{21}=\alpha-2q_{1}p_{2}-4q_{1}^{3}+q_{2}^{2}+2\lambda\big{(}p_{2}+q_{1}^{2}\big{)}+8q_{1}\lambda^{2}-16\lambda^{3}$.
The characteristic equation is

$\displaystyle z^{2}-256\lambda^{5}+16\alpha\lambda^{2}+8\lambda
h^{(q)}-f^{(q)}=0,$

with $h^{(q)}$ and $f^{(q)}$ given by (3.10) and (3.15b). We have

$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(q)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(q)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$

with

$\displaystyle U=\left(\begin{matrix}0&1\\\ \lambda-
q_{1}&0\end{matrix}\right),\qquad
L^{(1)}=\left(\begin{matrix}2q_{2}&-4(2\lambda+q_{1})\\\
2\big{(}p_{2}+2q_{1}^{2}+2\lambda
q_{1}-4\lambda^{2}\big{)}&-2q_{2}\end{matrix}\right),$

where $L^{(1)}$ is derived from $V^{(1)}$ of (3.17).

Using the coordinates of (3.12b), we define

$\displaystyle
L^{(2)}=8V^{(2)}=\left(\begin{matrix}\displaystyle\sum_{2}^{N}Q_{i}P_{i}-4\lambda
P_{1}&16\lambda^{2}+8\lambda Q_{1}-{\bf Q}^{2}\\\ a_{21}&\displaystyle
4\lambda P_{1}-\sum_{2}^{N}Q_{i}P_{i}\end{matrix}\right),$ (3.19a) where
$\displaystyle{\bf Q}^{2}=\sum_{2}^{N}Q_{i}^{2},\qquad
a_{21}=16\lambda^{3}-8Q_{1}\lambda^{2}+\lambda\left(4Q_{1}^{2}+\sum_{2}^{N}Q_{i}^{2}\right)+\sum_{2}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right).$
The characteristic equation is $\displaystyle
z^{2}=256\lambda^{5}+32h^{(Q)}\lambda^{2}-8f^{(Q)}\lambda-2\Phi^{(Q)},$
(3.19b) where $f^{(Q)}$ and $\Phi^{(Q)}$ are given in (3.12c). We have
$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(Q)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(Q)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$
$\displaystyle\big{\\{}L^{(2)},h_{ij}\big{\\}}=0,$ (3.19c) for $2\leq i<j\leq
N$, and where $\displaystyle U=\left(\begin{matrix}0&1\\\ \lambda-
Q_{1}&0\end{matrix}\right),\qquad
L^{(1)}=\left(\begin{matrix}P_{1}&-2(2\lambda+Q_{1})\\\ -4\lambda^{2}+2\lambda
Q_{1}-\frac{1}{2}\big{(}2Q_{1}^{2}+{\bf
Q}^{2}\big{)}&-P_{1}\end{matrix}\right).$ (3.19d)

##  4 DWW stationary flows

We can follow the approach of Section 3 in the context of the DWW hierarchy.
We have three representations of ${\bf u}_{t_{n}}$, given by (2.5b), with a
stationary flow being the reduction to a finite-dimensional space with ${\bf
u}_{t_{n}}=0$. The “time” variable for this flow is $x$. In the KdV case the
Casimir of $B_{0}$ was just $H_{0}$, which gave us the $\alpha$ term in (3.1),
whilst, for the DWW operator $B_{0}$, we have two Casimirs, $H_{0}$, $H_{1}$.
The DWW operator $B_{1}$ has one Casimir $H_{0}$, corresponding to the first
order part, but we also have the squared eigenfunctions for $J_{0}$. We leave
the operator $B_{2}$ to a later discussion of the Miura maps in Section 5, so
now consider the two operators $B_{0}$ and $B_{1}$:

1\. Using $B_{0}$, we have

$\displaystyle
B_{0}\delta_{u}H_{n+2}=0\quad\Rightarrow\quad\delta_{u}(H_{n+2}-\alpha_{0}H_{0}-4\alpha_{1}H_{1})=0,$
(4.1a) giving the Lagrangian
$\displaystyle{\mathcal{L}}_{n+2}=H_{n+2}-\alpha_{0}u_{1}-\alpha_{1}\big{(}4u_{0}+u_{1}^{2}\big{)}.$
(4.1b)

Again, we use the (generalised) Legendre transformation to find canonical
coordinates $(q_{i},p_{i})$ and the Hamiltonian function.

2\. Using $B_{1}$, we have

$\displaystyle\delta_{u_{0}}H_{n+1}=\frac{1}{2}\varphi^{2},\qquad\delta_{u_{1}}H_{n+1}=-\alpha$
$\displaystyle\qquad{}\Rightarrow\quad{\mathcal{L}}_{n+1}=H_{n+1}+\alpha
u_{1}+\frac{1}{2}\varphi_{x}^{2}-\frac{1}{2}u_{0}\varphi^{2}-\frac{\beta}{\varphi^{2}}.$
(4.2)

Again, the extensions (3.3b) and (3.3e) can be introduced here.

###  4.1 The $\boldsymbol{t_{1}}$ flow

We build 3 different Hamiltonians.

The Lagrangian (4.1b), using $8H_{3}$ and labelling $-{\mathcal{L}}_{3}$ as
$\mathcal{L}$, gives

$\displaystyle{\mathcal{L}}=\frac{1}{2}u_{1x}^{2}-\frac{5}{8}u_{1}^{4}+\alpha_{0}u_{1}+\alpha_{1}u_{1}^{2}+u_{0}\big{(}4\alpha_{1}-3u_{1}^{2}\big{)}-2u_{0}^{2}.$
(4.3a) This Lagrangian is degenerate, but $\delta_{u_{0}}{\mathcal{L}}=0$
gives $u_{0}=\alpha_{1}-\frac{3}{4}u_{1}^{2}$, after which
$\displaystyle{\mathcal{L}}=\frac{1}{2}u_{1x}^{2}+\frac{1}{2}u_{1}^{4}-2\alpha_{1}u_{1}^{2}+\alpha_{0}u_{1}+2\alpha_{1}^{2}$
$\displaystyle\qquad{}\Rightarrow\quad
h^{(q)}=\frac{1}{2}p_{1}^{2}-\frac{1}{2}q_{1}^{4}+2\alpha_{1}q_{1}^{2}-\alpha_{0}q_{1}.$
(4.3b) The fluxes (2.5e) give $\displaystyle F_{10}=2\alpha_{1},\qquad
F_{11}=\frac{1}{2}\alpha_{0},\qquad
F_{12}=-\frac{1}{4}h^{(q)}+\frac{1}{2}\alpha_{1}^{2}.$ (4.3c)

The Lagrangian (4.2), using $-H_{2}$ in the first multicomponent version of
(4.2) (see (3.3b)), we find

$\displaystyle\tilde{\mathcal{L}}=\sum_{i=1}^{N}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{\beta_{i}}{\varphi_{i}^{2}}\right)+\alpha
u_{1}-\frac{1}{8}u_{1}\left(4u_{0}+u_{1}^{2}\right).$ (4.4a) This is also
degenerate, but $\delta_{u_{0}}\tilde{\mathcal{L}}=0$ and
$\delta_{u_{1}}\tilde{\mathcal{L}}=0$ give $u_{1}=-\sum\varphi_{i}^{2}$ and
$u_{0}=2\alpha-\frac{3}{4}\big{(}\sum\varphi_{i}^{2}\big{)}^{2}$, after which
$\displaystyle\tilde{\mathcal{L}}=\sum_{i=1}^{N}\left(\frac{1}{2}\varphi_{ix}^{2}-\alpha\varphi_{i}^{2}-\frac{\beta_{i}}{\varphi_{i}^{2}}\right)+\frac{1}{8}\left(\sum_{i=1}^{N}\varphi_{i}^{2}\right)^{3}.$
(4.4b) Defining $Q_{i}=\varphi_{i}$, $P_{i}=\varphi_{ix}$, we obtain the
Hamiltonian $\displaystyle
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}\left(P_{i}^{2}+2\alpha
Q_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)-\frac{1}{8}\left(\sum_{i=1}^{N}Q_{i}^{2}\right)^{3}.$
(4.4c) The fluxes (2.5e) give (compare with (3.6b)) $\displaystyle
F_{10}=4\alpha,\qquad F_{11}=-2h^{(Q)},\qquad
F_{12}-2\alpha^{2}=\Phi^{(Q)}=\frac{1}{2}\sum_{1\leq i<j\leq
N}h_{ij}+\sum_{i=1}^{N}\beta_{i}.$ (4.4d)

Since $h_{ij}$ are a deformation of the (squares of) angular momenta, it is
straightforward to build an involutive set of integrals, so (4.4c) defines a
superintegrable system (for $N\geq 3$).

When $N=1$, $F_{12}=2\alpha^{2}+\beta$. When $N=2$, we have 2 integrals
$h^{(Q)}$, $h_{12}$. When $N=3$, we have 4 integrals $h^{(Q)}$, $h_{ij}$, with
$h^{(Q)}$, $h_{12}$ and $\Phi^{(Q)}$ in involution. This and the Hamiltonian
(3.5b) (for $N=3$) are particular cases of the first potential in Table II of
[11].

######  Remark 4.1 (rotationally invariant case).

The extension (3.3e) gives a similar result, with

$\displaystyle h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}\left(P_{i}^{2}+2\alpha
Q_{i}^{2}\right)-\frac{1}{8}\left(\sum_{i=1}^{N}Q_{i}^{2}\right)^{3}+\frac{\beta}{{\bf
Q}^{2}},$ (4.5)

where ${\bf Q}^{2}=\sum_{i=1}^{N}Q_{i}^{2}$. The integrals $h_{ij}$ are just
replaced by the rotation algebra.

####  4.1.1 Bi-Hamiltonian formulation when $\boldsymbol{N=1}$

Here we have

$\displaystyle h^{(Q)}=\frac{1}{2}P_{1}^{2}-\frac{1}{8}Q_{1}^{6}+\alpha
Q_{1}^{2}+\frac{\beta}{Q_{1}^{2}}.$ (4.6)

The Hamiltonian formulations (4.3b) and (4.6) give coordinates ${\bf
q}=(q_{1},p_{1},\alpha_{0},\alpha_{1})$ and ${\bf
Q}=(Q_{1},P_{1},\beta,\alpha)$, respectively. From the definitions of the
canonical coordinates, we have

$\displaystyle q_{1}=-Q_{1}^{2},\qquad
p_{1}=-2Q_{1}P_{1},\qquad\alpha_{0}=-4h^{(Q)},\qquad\alpha_{1}=2\alpha.$

Following the same procedure as Section 3.1.1, we introduce a degenerate
extension of the canonical Poisson tensor in the 4D space with coordinates
$\bf q$ and then define $\big{(}\frac{\partial{\bf Q}}{\partial{\bf
q}}\big{)}{\mathcal{P}}_{0}^{(q)}\big{(}\frac{\partial{\bf Q}}{\partial{\bf
q}}\big{)}^{\rm T}=\frac{1}{4}{\mathcal{P}}_{0}^{(Q)}$, giving

$\displaystyle{\mathcal{P}}_{0}^{(q)}=\left(\begin{matrix}\phantom{-}0&1&0&0\\\
-1&0&0&0\\\ \phantom{-}0&0&0&0\\\
\phantom{-}0&0&0&0\end{matrix}\right)\qquad\text{and}\qquad{\mathcal{P}}_{0}^{(Q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}\frac{1}{Q_{1}^{2}}&a_{13}&0\\\
-\frac{1}{Q_{1}^{2}}&\phantom{-}0&a_{23}&0\\\ -a_{13}&-a_{23}&0&0\\\
\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$

where $(a_{13},a_{23},0,0)^{\rm T}=-P_{1}^{(Q)}\nabla_{Q}h^{(Q)}$ and
${\mathcal{P}}_{1}^{(Q)}$ is the compatible (degenerate) canonical Poisson
tensor on the $\bf Q$ space, which in turn leads to $\big{(}\frac{\partial{\bf
q}}{\partial{\bf Q}}\big{)}{\mathcal{P}}_{1}^{(Q)}\big{(}\frac{\partial{\bf
q}}{\partial{\bf Q}}\big{)}^{\rm T}=-4{\mathcal{P}}_{1}^{(q)}$, giving

$\displaystyle{\mathcal{P}}_{1}^{(Q)}=\left(\begin{matrix}\phantom{-}0&1&0&0\\\
-1&0&0&0\\\ \phantom{-}0&0&0&0\\\
\phantom{-}0&0&0&0\end{matrix}\right)\qquad\text{and}\qquad{\mathcal{P}}_{1}^{(q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}q_{1}&b_{13}&0\\\
-q_{1}&\phantom{-}0&b_{23}&0\\\ -b_{13}&-b_{23}&0&0\\\
\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$

where $(b_{13},b_{23},0,0)^{\rm T}=P_{0}^{(q)}\nabla_{q}h^{(q)}$.

We use the three independent fluxes from (2.5e) to build the bi-Hamiltonian
ladders:

$\displaystyle F_{10}=2\alpha_{1}=4\alpha,\qquad
F_{11}=\frac{1}{2}\alpha_{0}=-2h^{(Q)},\qquad
F_{12}=-\frac{1}{4}h^{(q)}+\frac{1}{2}\alpha_{1}^{2}=2\alpha^{2}+\beta,$

which satisfy
${\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\alpha={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\alpha=0$
and

$\displaystyle{\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}=0,\qquad{\bf
q}_{t_{h}}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{0}={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},\qquad{\mathcal{P}}_{1}^{(q)}\nabla_{q}h^{(q)}=0,$
$\displaystyle{\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0,\qquad{\bf
Q}_{t_{h}}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}(-\beta),\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta=0.$

######  Remark 4.2 (the parameters $\alpha$ and $\alpha_{1}$).

At this stage it looks like we should consider $\alpha_{1}$ and $\alpha$ as
_constants_ rather than dynamical variables. However, we will see in Section 5
that they generate nontrivial flows with respect to a _third_ Poisson bracket.

###  4.2 The $\boldsymbol{t_{2}}$ flow

We build 3 different Hamiltonians.

The Lagrangian (4.1b), using $2H_{4}$ and labelling ${\mathcal{L}}_{4}$ as
$\mathcal{L}$, gives

$\displaystyle{\mathcal{L}}=\frac{1}{64}\big{(}u_{1}\big{(}4u_{0}+u_{1}^{2}\big{)}\big{(}12u_{0}+7u_{1}^{2}\big{)}-4u_{1x}(4u_{0x}+5u_{1}u_{1x})\big{)}-\alpha_{0}u_{1}-\alpha_{1}\big{(}4u_{0}+u_{1}^{2}\big{)}.\\!\\!\\!$
(4.7a) The standard Legendre transformation gives $u_{0}=q_{1},\qquad
u_{1}=q_{2},\qquad u_{0x}=2(5q_{2}p_{1}-2p_{2}),\qquad u_{1x}=-4p_{1},$ giving
the Hamiltonian $\displaystyle
h^{(q)}=5q_{2}p_{1}^{2}-4p_{1}p_{2}-\frac{q_{2}}{64}\big{(}4q_{1}+q_{2}^{2}\big{)}\big{(}12q_{1}+7q_{2}^{2}\big{)}+\alpha_{0}q_{2}+\alpha_{1}\big{(}4q_{1}+q_{2}^{2}\big{)}.$
(4.7b) The fluxes (2.5e) give $\displaystyle F_{20}=8\alpha_{1},\qquad
F_{21}=2\alpha_{0},\qquad F_{22}=h^{(q)},$ (4.7c) $\displaystyle
F_{23}-8\alpha_{1}^{2}=f^{(q)}=\frac{1}{256}\big{(}4q_{1}+3q_{2}^{2}\big{)}\big{(}64(\alpha_{0}-2\alpha_{1}q_{2})+\\!64p_{1}^{2}-16q_{1}^{2}+5q_{2}^{4}\big{)}-2(p_{2}-q_{2}p_{1})^{2}.$

The Lagrangian (4.2), using $-8H_{3}$ in the first multicomponent version of
(4.2) (see (3.3b)), we find

$\displaystyle\tilde{\mathcal{L}}=\sum_{i=1}^{N-1}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{\beta_{i+1}}{\varphi_{i}^{2}}\right)+\alpha
u_{1}+\frac{1}{2}u_{1x}^{2}-\frac{1}{8}\big{(}4u_{0}+u_{1}^{2}\big{)}\big{(}4u_{0}+5u_{1}^{2}\big{)}.$
This is degenerate in $u_{0}$, but $\delta_{u_{0}}\tilde{\mathcal{L}}=0$ gives
$u_{0}=-\frac{3}{4}u_{1}^{2}-\frac{1}{8}\sum_{1}^{N-1}\varphi_{i}^{2}$, after
which
$\displaystyle\tilde{\mathcal{L}}=\frac{1}{2}\left(u_{1x}^{2}+\sum_{i=1}^{N-1}\left(\varphi_{ix}^{2}-\frac{2\beta_{i+1}}{\varphi_{i}^{2}}\right)\right)+\alpha
u_{1}+\frac{1}{32}\left(16u_{1}^{4}+12u_{1}^{2}\sum_{i=1}^{N-1}\varphi_{i}^{2}+\left(\sum_{i=1}^{N-1}\varphi_{i}^{2}\right)^{2}\right).$
Defining $u_{1}=Q_{1}$, $u_{1x}=P_{1}$ and $\varphi_{i}=Q_{i+1}$,
$\varphi_{ix}=P_{i+1}$, $i=1,\dots,N-1$, we obtain the Hamiltonian
$\displaystyle h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}-\alpha
Q_{1}+\sum_{i=2}^{N}\frac{\beta_{i}}{Q_{i}^{2}}-\frac{1}{32}\left(16Q_{1}^{4}+12Q_{1}^{2}\sum_{i=2}^{N}Q_{i}^{2}+\left(\sum_{i=2}^{N}Q_{i}^{2}\right)^{2}\right).$
(4.8a) The fluxes (2.5e) give $\displaystyle F_{20}=\frac{1}{2}\alpha,\qquad
F_{21}=-\frac{1}{4}h^{(Q)},\qquad
64F_{23}-2\alpha^{2}=\Phi^{(Q)}=\frac{1}{2}\sum_{2\leq i<j\leq
N}h_{ij}+\sum_{i=2}^{N}\beta_{i},$ (4.8b) $\displaystyle
16F_{22}=f^{(Q)}=P_{1}\sum_{i}^{N}Q_{i}P_{i}-2Q_{1}h^{(Q)}-\frac{1}{2}\alpha\\!\big{(}4Q_{1}^{2}+{\bf
Q}^{2}\big{)}-\frac{1}{16}Q_{1}\big{(}4Q_{1}^{2}+{\bf
Q}^{2}\big{)}\big{(}4Q_{1}^{2}+3{\bf Q}^{2}\big{)},$ where ${\bf
Q}^{2}=\sum_{2}^{N}Q_{i}^{2}$ and $h_{ij}$ are defined by (3.6a), with $2\leq
i<j\leq N$.

In fact, $h_{ij}$ are themselves first integrals. Since $h_{ij}$ are a
deformation of the (squares of) angular momenta, it is straightforward to
build an involutive set of integrals, so (4.8a) defines a superintegrable
system (for $N\geq 4$).

When $N=2$, $F_{23}=\frac{1}{64}\big{(}2\alpha^{2}+\beta\big{)}$, so there are
just the two integrals, $h^{(Q)}$ and $f^{(Q)}$. This is the case which was
found in the classifications of [14, 19]. We show that this particular case is
tri-Hamiltonian in Section 5 (see Remark 5.3). In fact, this Hamiltonian and
the Hénon–Heiles one (3.13) belong to an infinite family, which are separable
in parabolic coordinates (see, for example, [18, equation (2.2.41)]). Other
members of the classifications of [14, 19] can similarly be derived as
stationary flows associated with $4^{\rm th}$ order scalar Lax operators [4].

When $N=3$, we have $h^{(Q)}$, $f^{(Q)}$ and $\Phi^{(Q)}$ in involution. It
appears in the classification of [10, case 4 of Table 1] and is further
generalised in [13] and in Section 6 of this paper.

When $N=4$, we have $h^{(Q)}$, $f^{(Q)}$, $h_{23}$ and $\Phi^{(Q)}$ in
involution, with five independent integrals.

######  Remark 4.3 (rotationally invariant case).

The extension (3.3e) gives a similar result, with

$\displaystyle h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}-\alpha
Q_{1}-\frac{1}{32}\left(16Q_{1}^{4}+12Q_{1}^{2}\sum_{i=2}^{N}Q_{i}^{2}+\left(\sum_{i=2}^{N}Q_{i}^{2}\right)^{2}\right)+\frac{\beta}{{\bf
Q}^{2}}.$ (4.9)

The integrals $h_{ij}$ are just replaced by the rotation algebra on the space
$Q_{2},\dots,Q_{N}$.

####  4.2.1 Bi-Hamiltonian formulation when $\boldsymbol{N=2}$

Here we have

$\displaystyle h^{(Q)}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}\big{)}-\alpha
Q_{1}-\frac{1}{32}\big{(}16Q_{1}^{4}+12Q_{1}^{2}Q_{2}^{2}+Q_{2}^{4}\big{)}+\frac{\beta}{Q_{2}^{2}}.$
(4.10)

The Hamiltonian formulations (4.7b) and (4.10) give coordinates ${\bf
q}=(q_{i},p_{i},\alpha_{0},\alpha_{1})$ and ${\bf
Q}=(Q_{i},P_{i},\beta,\alpha)$, respectively. From the definitions of the
canonical coordinates, we have

$\displaystyle q_{1}=-\frac{1}{8}\big{(}6Q_{1}^{2}+Q_{2}^{2}\big{)},\qquad
q_{2}=Q_{1},\qquad p_{1}=-\frac{1}{4}P_{1},$ $\displaystyle
p_{2}=\frac{1}{16}(Q_{2}P_{2}-4Q_{1}P_{1}),\qquad\alpha_{0}=-\frac{1}{8}h^{(Q)},\qquad\alpha_{1}=\frac{1}{16}\alpha.$

Following the same procedure as Section 3.1.1, we introduce a degenerate
extension of the canonical Poisson tensor in the 6D space with coordinates
$\bf q$ and then use $\big{(}\frac{\partial{\bf Q}}{\partial{\bf
q}}\big{)}{\mathcal{P}}_{0}^{(q)}\big{(}\frac{\partial{\bf Q}}{\partial{\bf
q}}\big{)}^{\rm T}=16{\mathcal{P}}_{0}^{(Q)}$, giving

$\displaystyle{\mathcal{P}}_{0}^{(q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&1&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&1&0&0\\\ -1&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&-1&0&0&0&0\\\ \phantom{-}0&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&0&0&0\end{matrix}\right),$
$\displaystyle{\mathcal{P}}_{0}^{(Q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{Q_{2}}&a_{15}&0\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{Q_{2}}&-\frac{2Q_{1}}{Q_{2}^{2}}&a_{25}&0\\\\[5.69054pt]
\phantom{-}0&-\frac{1}{Q_{2}}&\phantom{-}0&\phantom{-}\frac{P_{2}}{Q_{2}^{2}}&a_{35}&0\\\\[5.69054pt]
-\frac{1}{Q_{2}}&\phantom{-}\frac{2Q_{1}}{Q_{2}^{2}}&-\frac{P_{2}}{Q_{2}^{2}}&\phantom{-}0&a_{45}&0\\\\[5.69054pt]
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$
(4.11a) where the column $(a_{15},a_{25},a_{35},a_{45},0,0)^{\rm
T}=-{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}$, where $P_{1}^{(Q)}$ is the
compatible (degenerate) canonical Poisson tensor on the $\bf Q$ space, which
in turn leads to $\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}{\mathcal{P}}_{1}^{(Q)}\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}^{\rm T}=\frac{1}{32}{\mathcal{P}}_{1}^{(q)}$:
$\displaystyle{\mathcal{P}}_{1}^{(Q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&1&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&1&0&0\\\ -1&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&-1&0&0&0&0\\\ \phantom{-}0&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&0&0&0\end{matrix}\right),$
$\displaystyle{\mathcal{P}}_{1}^{(q)}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&12q_{2}&\phantom{-}b_{14}&b_{15}&0\\\\[2.84526pt]
\phantom{-}0&\phantom{-}0&-8&-8q_{2}&b_{25}&0\\\\[2.84526pt]
-12q_{2}&\phantom{-}8&\phantom{-}0&\phantom{-}8p_{1}&b_{35}&0\\\\[2.84526pt]
-b_{14}&\phantom{-}8q_{2}&-8p_{1}&\phantom{-}0&b_{45}&0\\\\[2.84526pt]
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0&0\\\\[2.84526pt]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$
(4.11b) where $b_{14}=4q_{1}+15q_{2}^{2}$ and the column
$(b_{15},b_{25},b_{35},b_{45},0,0)^{\rm
T}=-4{\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}$.

We use the independent fluxes from (2.5e) to build the bi-Hamiltonian ladders:

$\displaystyle F_{20}=8\alpha_{1}=\frac{1}{2}\alpha,\qquad
F_{21}=2\alpha_{0}=-\frac{1}{4}h^{(Q)},$ $\displaystyle
F_{22}=h^{(q)}=\frac{1}{16}f^{(Q)},\qquad F_{23}=f^{(q)}=\frac{\beta}{64},$

where $f^{(q)}$ is given by (4.7c) and $f^{(Q)}$ by (4.8b) (with $N=2$). These
satisfy

${\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}=0\qquad\text{and}\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\alpha={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\alpha=0,$

together with

$\displaystyle{\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}=0,\qquad{\bf
q}_{t_{h}}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{4}\alpha_{0}\right)={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},$
$\displaystyle{\bf
q}_{t_{f}}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{8}h^{(q)}\right)=P_{0}^{(q)}f^{(q)},\quad{\mathcal{P}}_{1}^{(q)}\nabla_{q}f^{(q)}=0,$
$\displaystyle{\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0,\qquad{\bf
Q}_{t_{h}}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}f^{(Q)},$
$\displaystyle{\bf
Q}_{t_{f}}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}(-\beta),\qquad{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta=0.$

######  Remark 4.4.

The first two components of the $t_{h}$ and $t_{f}$ flows just reproduce the
first two flows of the DWW hierarchy, analogous to equations (3.16a) and
(3.16b) for the stationary $t_{2}$ flow of the KdV hierarchy.

###  4.3 The Lax representation

Following the approach of Section 3.3, we obtain the Lax representations.

####  4.3.1 The $\boldsymbol{t_{1}}$ flow

In this case our matrices are

$\displaystyle U=\left(\begin{matrix}0&1\\\ \lambda^{2}-\lambda
u_{1}-u_{0}&0\end{matrix}\right),\qquad
V^{(1)}=\left(\begin{matrix}-\frac{1}{2}u_{1x}&2\lambda+u_{1}\\\
a_{21}&\frac{1}{2}u_{1x}\end{matrix}\right).$ (4.12)

where $a_{21}=\big{(}\lambda^{2}-\lambda
u_{1}-u_{0}\big{)}(2\lambda+u_{1})-\frac{1}{2}a_{12xx}$, given in terms of the
component $a_{12}$ of the matrix $V^{(1)}$.

Using the coordinates of the Hamiltonian (4.3b), we define

$\displaystyle L^{(1)}=-2V^{(1)}=\begin{pmatrix}p_{1}&-2(2\lambda+q_{1})\\\
a_{21}&-p_{1}\end{pmatrix}\quad\Rightarrow\quad
z^{2}-2h^{(q)}+4\alpha_{0}\lambda+16\alpha_{1}\lambda^{2}-16\lambda^{4}=0,$

where
$a_{21}=\alpha_{0}-2\alpha_{1}q_{1}+\frac{1}{2}q_{1}^{3}+\lambda\big{(}4\alpha_{1}-q_{1}^{2}\big{)}+2\lambda^{2}q_{1}-4\lambda^{3}$.
The Lax representation of the equation generated by $h^{(q)}$ is

$\displaystyle
L^{(1)}_{x}=\big{\\{}L^{(1)},h^{(q)}\big{\\}}=\big{[}U,L^{(1)}\big{]},\qquad\text{with}\quad
U=\left(\begin{matrix}0&1\\\ \lambda^{2}-\lambda
q_{1}+\frac{3}{4}q_{1}^{2}-\alpha_{1}&0\end{matrix}\right).$

Using the coordinates of (4.4c), we have

$\displaystyle
L^{(1)}=V^{(1)}=\left(\begin{matrix}\displaystyle\sum_{1}^{N}Q_{i}P_{i}&2\lambda-{\bf
Q}^{2}\\\ a_{21}&\displaystyle-\sum_{1}^{N}Q_{i}P_{i}\end{matrix}\right)$
$\displaystyle\quad{}\Rightarrow\quad z^{2}+2\Phi^{(Q)}-4\lambda
h^{(Q)}+8\alpha\lambda^{2}-4\lambda^{4}=0,$ (4.13a) where
$a_{21}=\sum_{1}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)+\frac{1}{2}\lambda\big{(}{\bf
Q}^{4}-8\alpha\big{)}+\lambda^{2}{\bf Q}^{2}+2\lambda^{3},\qquad{\bf
Q}^{2}=\sum_{1}^{N}Q_{i}^{2},\quad{\bf Q}^{4}=\left({\bf Q}^{2}\right)^{2},$
with $\Phi^{(Q)}$ as given in (4.4d). The Lax equations generated by $h^{(Q)}$
and $\Phi^{(Q)}$ are $\displaystyle
L^{(1)}_{x}=\big{\\{}L^{(1)},h^{(Q)}\big{\\}}=\big{[}U,L^{(1)}\big{]},$
$\displaystyle\big{\\{}L^{(1)},\Phi^{(Q)}\big{\\}}=0,\quad\text{with}\quad
U=\begin{pmatrix}0&1\\\ \lambda^{2}+\lambda{\bf Q}^{2}+\frac{3}{4}{\bf
Q}^{4}-2\alpha&0\end{pmatrix}.$ (4.13b)

In fact, $\big{\\{}L^{(1)},h_{ij}\big{\\}}=0$, for each of the functions
$h_{ij}$ of (3.6a).

######  Remark 4.5 (rotationally symmetric case).

Similar equations hold for the case (4.5).

####  4.3.2 The $\boldsymbol{t_{2}}$ flow

In this case our matrices are $U$ (the same as (4.12)) and

$\displaystyle V^{(2)}=\left(\begin{matrix}-\frac{1}{2}\lambda
u_{1x}-\frac{1}{4}(2u_{0x}+3u_{1}u_{1x})&2\lambda^{2}+\lambda
u_{1}+u_{0}+\frac{3}{4}u_{1}^{2}\\\ a_{21}&\frac{1}{2}\lambda
u_{1x}+\frac{1}{4}(2u_{0x}+3u_{1}u_{1x})\end{matrix}\right),$

where

$a_{21}=\left(2\lambda^{2}+\lambda
u_{1}+u_{0}+\frac{3}{4}u_{1}^{2}\right)\big{(}\lambda^{2}-\lambda
u_{1}-u_{0}\big{)}-\frac{1}{2}a_{12xx}.$

Using the coordinates of the Hamiltonian (4.7b), we define

$\displaystyle L^{(2)}=\frac{1}{2}V^{(2)}=\left(\begin{matrix}\lambda
p_{1}+p_{2}-q_{2}p_{1}&\lambda^{2}+\frac{1}{2}q_{2}\lambda+\frac{1}{8}\big{(}4q_{1}+3q_{2}^{2}\big{)}\\\
a_{21}&q_{2}p_{1}-p_{2}-\lambda p_{1}\end{matrix}\right),$

where

$\displaystyle
a_{21}=\lambda^{4}-\frac{1}{2}q_{2}\lambda^{3}-\frac{1}{8}\big{(}4q_{1}+q_{2}^{2}\big{)}\lambda^{2}+\frac{1}{4}\big{(}q_{2}^{3}+2q_{1}q_{2}-16\alpha_{1}\big{)}\lambda-\alpha_{0}$
$\displaystyle\hphantom{a_{21}=}{}+2\alpha_{1}q_{2}+\frac{1}{4}q_{1}^{2}-\frac{5}{64}q_{2}^{4}-p_{1}^{2}.$

The characteristic equation of $L^{(2)}$ is

$\displaystyle
z^{2}+\frac{1}{2}f^{(q)}+\frac{1}{2}h^{(q)}\lambda+\alpha_{0}\lambda^{2}+4\alpha_{1}\lambda^{3}-\lambda^{6}=0,$

where $f^{(q)}$ is given by (4.7c).

The Lax representation of the equations generated by $h^{(q)}$ and $f^{(q)}$
are

$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(q)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(q)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$

where $U$ and $L^{(1)}=\frac{1}{2}V^{(1)}$ (see (4.12)) are given by

$\displaystyle U=\left(\begin{matrix}0&1\\\ \lambda^{2}-q_{2}\lambda-
q_{1}&0\end{matrix}\right),\qquad
L^{(1)}=\left(\begin{matrix}p_{1}&\lambda+\frac{1}{2}q_{2}\\\
a_{21}&-p_{1}\end{matrix}\right),$

with

$a_{21}=\lambda^{3}-\frac{1}{2}q_{2}\lambda^{2}-\frac{1}{2}\big{(}2q_{1}+q_{2}^{2}\big{)}\lambda+\frac{1}{8}\big{(}5q_{2}^{3}+8q_{1}q_{2}-32\alpha_{1}\big{)}.$

Using the coordinates of (4.8a), we have

$\displaystyle
L^{(2)}=8V^{(2)}=\left(\begin{matrix}\displaystyle\sum_{2}^{N}Q_{i}P_{i}-4\lambda
P_{1}&16\lambda^{2}+8\lambda Q_{1}-\displaystyle\sum_{2}^{N}Q_{i}^{2}\\\
a_{21}&4\lambda P_{1}-\displaystyle\sum_{2}^{N}Q_{i}P_{i}\end{matrix}\right),$
(4.14a) where $\displaystyle
a_{21}=\sum_{2}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)-\lambda\left(4\alpha+Q_{1}\left(2Q_{1}^{2}+\sum_{2}^{N}Q_{i}^{2}\right)\right)$
$\displaystyle\hphantom{a_{21}=}{}+\lambda^{2}\left(4Q_{1}^{2}+\sum_{2}^{N}Q_{i}^{2}\right)-8Q_{1}\lambda^{3}+16\lambda^{4}.$
The characteristic equation of $L^{(2)}$ is $\displaystyle
z^{2}+2\Phi^{(Q)}+8f^{(Q)}\lambda-32h^{(Q)}\lambda^{2}+64\alpha\lambda^{3}-256\lambda^{6}=0,$
(4.14b) where $\Phi^{(Q)}$ and $f^{(Q)}$ are given in (4.8b).

The Lax equations generated by $h^{(Q)}$, $f^{(Q)}$ and $h_{ij}$ (not just the
combination $\Phi^{(Q)}$) are

$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(Q)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(Q)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$
$\displaystyle\big{\\{}L^{(2)},h_{ij}\big{\\}}=0,$ (4.14c)

where $L^{(1)}=-2V^{(1)}$ (see (4.12)), written in the coordinates of (4.8a),
and $U$ are given by

$\displaystyle L^{(1)}=\left(\begin{matrix}P_{1}&-4\lambda-2Q_{1}\\\
a_{21}&-P_{1}\end{matrix}\right),\qquad U=\left(\begin{matrix}0&1\\\
\lambda^{2}-\lambda
Q_{1}+\frac{1}{8}\left(6Q_{1}^{2}+\sum_{2}^{N}Q_{i}^{2}\right)&0\end{matrix}\right),$
(4.14d)

with

$a_{21}=\alpha+\frac{1}{2}Q_{1}\sum_{1}^{N}Q_{i}^{2}-\frac{1}{2}\lambda\left(2Q_{1}^{2}+\sum_{2}^{N}Q_{i}^{2}\right)+2Q_{1}\lambda^{2}-4\lambda^{3}.$

######  Remark 4.6 (rotationally symmetric case).

Similar equations hold for the case (4.9).

##  5 The DWW stationary flows in tri-Hamiltonian form

We now use the Miura maps of Section 2.2.1 to construct a tri-Hamiltonian
formulation of the stationary flows corresponding to (2.5b) and (2.7). In this
way, Figure 1 is extended to an array of 9 Poisson matrices
${\mathcal{P}}_{k}^{(m)}$, $k,m=0,1,2$. The 3 Poisson matrices
${\mathcal{P}}_{k}^{(k)}$ can be directly constructed in canonical form,
giving us coordinates $(q_{i},p_{i})$, $(Q_{i},P_{i})$ and
$\big{(}\bar{Q}_{i},\bar{P}_{i}\big{)}$, for $k=0,1,2$ respectively. Written
in terms of these coordinates, the Miura maps then allow us to build the
remaining 6 Poisson matrices.

  1. 1.

The case $B_{0}^{u}$, has already been considered in Section 4, giving the
Lagrangian (4.1b), which we now write as

$\displaystyle{\mathcal{L}}_{n+2}^{u}=H_{n+2}^{u}-\alpha_{0}u_{1}-\alpha_{1}\big{(}4u_{0}+u_{1}^{2}\big{)}.$
(5.1)

The (generalised) Legendre transformation defines canonical coordinates
$(q_{i},p_{i})$, the Poisson matrix
${\mathcal{P}}_{0}^{(0)}\equiv{\mathcal{P}}_{0}^{(q)}$ and the Hamiltonian
function $h^{(q)}$.

  2. 2.

Using $B_{1}^{w}$, we have

$\displaystyle
B_{1}^{w}\delta_{w}H_{n+1}^{w}=0\quad\Rightarrow\quad\delta_{w}\left(H_{n+1}^{w}-\beta_{0}w_{0}-\beta_{1}w_{1}\right)=0$
$\displaystyle\qquad{}\Rightarrow\quad{\mathcal{L}}_{n+1}^{w}=H_{n+1}^{w}-\beta_{0}w_{0}-\beta_{1}w_{1}.$
(5.2)

The (generalised) Legendre transformation defines canonical coordinates
$(Q_{i},P_{i})$, the Poisson matrix
${\mathcal{P}}_{1}^{(1)}\equiv{\mathcal{P}}_{1}^{(Q)}$ and the Hamiltonian
function $h^{(Q)}$.

  3. 3.

Using $B_{2}^{v}$, we have

$\displaystyle
B_{2}^{v}\delta_{v}H_{n}^{v}=0\quad\Rightarrow\quad\delta_{v}\left(H_{n}^{v}-\gamma_{0}v_{0}-\gamma_{1}v_{1}\right)=0\quad\Rightarrow\quad{\mathcal{L}}_{n}^{v}=H_{n}^{v}-\gamma_{0}v_{0}-\gamma_{1}v_{1}.\\!\\!\\!$
(5.3)

The (generalised) Legendre transformation defines canonical coordinates
$\big{(}\bar{Q}_{i},\bar{P}_{i}\big{)}$, the Poisson matrix
${\mathcal{P}}_{2}^{(2)}\equiv{\mathcal{P}}_{2}^{(\bar{Q})}$ and the
Hamiltonian function $h^{(\bar{Q})}$.

###  5.1 The $\boldsymbol{t_{1}}$ flow

We now derive the explicit formulae for the case $n=1$.

The $u$ space, with (5.1): We previously derived this as (4.3b), giving

$\displaystyle
h^{(q)}=\frac{1}{2}p_{1}^{2}-\frac{1}{2}q_{1}^{4}+2\alpha_{1}q_{1}^{2}-\alpha_{0}q_{1},$
(5.4)

where $u_{0}=\alpha_{1}-\frac{3}{4}u_{1}^{2}$, $u_{1}=q_{1}$, $u_{1x}=p_{1}$.

The $w$ space, with (5.2): Using $-H_{2}^{w}$, we have

$\displaystyle{\mathcal{L}}_{2}^{w}=\frac{1}{2}w_{1}\big{(}w_{0x}+w_{0}^{2}\big{)}-\frac{1}{8}w_{1}^{3}-\beta_{0}w_{0}-\beta_{1}w_{1}.$
(5.5a) This is degenerate and reduces to
$w_{0}=\frac{w_{1x}+2\beta_{0}}{2w_{1}}$. Removing an exact derivative and an
overall numerical factor, we find
$\displaystyle{\mathcal{L}}_{2}^{w}=\frac{(w_{1x}+2\beta_{0})^{2}}{2w_{1}}+\frac{1}{2}w_{1}\big{(}w_{1}^{2}+8\beta_{1}\big{)},$
(5.5b) corresponding to the Hamiltonian $\displaystyle
h^{(Q)}=\frac{1}{2}Q_{1}P_{1}^{2}-2\beta_{0}P_{1}-\frac{1}{2}Q_{1}\big{(}8\beta_{1}+Q_{1}^{2}\big{)}.$
(5.5c)

The $v$ space, with (5.3): Using $H_{1}^{v}$, we have

$\displaystyle{\mathcal{L}}_{1}^{v}=-v_{0x}-v_{0}^{2}+\frac{1}{4}\big{(}2v_{0}v_{1}+v_{1x}\big{)}^{2}-\gamma_{0}v_{0}-\gamma_{1}v_{1}.$
(5.6a) This is degenerate and reduces to
$v_{0}=\frac{\gamma_{0}-v_{1}v_{1x}}{2(v_{1}^{2}-1)}$. Removing an exact
derivative and an overall numerical factor, we find
$\displaystyle{\mathcal{L}}_{1}^{v}=\frac{1}{2(1-v_{1}^{2})}\big{(}v_{1x}^{2}-2\gamma_{0}v_{1}v_{1x}-4\gamma_{1}v_{1}\big{(}1-v_{1}^{2}\big{)}+\gamma_{0}^{2}\big{)},$
(5.6b) corresponding to the Hamiltonian $\displaystyle
h^{(\bar{Q})}=\frac{1}{2}\big{(}1-\bar{Q}_{1}^{2}\big{)}\bar{P}_{1}^{2}+\gamma_{0}\bar{Q}_{1}\bar{P}_{1}+2\gamma_{1}\bar{Q}_{1}.$
(5.6c)

####  5.1.1 The Miura maps in these coordinates

We consider the two steps induced by ${\bf w}\mapsto{\bf u}$ and ${\bf
v}\mapsto{\bf w}$. We again extend each space to include the parameters as
dynamical variables, which we define as ${\bf
q}=(q_{1},p_{1},\alpha_{0},\alpha_{1})$, ${\bf
Q}=(Q_{1},P_{1},\beta_{0},\beta_{1})$,
${\bf\bar{Q}}=\big{(}\bar{Q}_{1},\bar{P}_{1},\gamma_{0},\gamma_{1}\big{)}$.

The relation of $\bf q$ to $\bf Q$: $q_{1}=Q_{1}$,
$p_{1}=Q_{1}P_{1}-2\beta_{0}$. The constraints on $u_{0}$ and $w_{0}$,
together with $u_{0}=-w_{0x}-w_{0}^{2}$, gives $\alpha_{1}=-2\beta_{1}$. The
formula $\alpha_{0}=u_{1xx}-2u_{1}^{3}+4\alpha_{1}u_{1}$ gives
$\alpha_{0}=h^{(Q)}$. In summary

$\displaystyle q_{1}=Q_{1},\qquad
p_{1}=Q_{1}P_{1}-2\beta_{0},\qquad\alpha_{0}=h^{(Q)},\qquad\alpha_{1}=-2\beta_{1}\qquad\Rightarrow\qquad
h^{(q)}=2\beta_{0}^{2}.$

The relation of $\bf Q$ to $\bf\bar{Q}$: $Q_{1}=-\bar{P}_{1}$,
$P_{1}=\bar{Q}_{1}\bar{P}_{1}-\gamma_{0}$. The “$x$ derivative” of the first
gives
$\big{\\{}Q_{1},h^{(Q)}\big{\\}}=\big{\\{}{-}\bar{P}_{1},h^{(\bar{Q})}\big{\\}}$,
which implies $\beta_{0}=-\gamma_{1}$. The formula
$\beta_{1}=\frac{4\beta_{0}P_{1}-Q_{1}(P_{1}^{2}+3Q_{1}^{2})+2Q_{1xx}}{8Q_{1}}$
gives $\beta_{1}=-\frac{1}{4}h^{(\bar{Q})}+\frac{1}{8}\gamma_{0}^{2}$. In
summary

$\displaystyle Q_{1}=-\bar{P}_{1},\qquad
P_{1}=\bar{Q}_{1}\bar{P}_{1}-\gamma_{0},\qquad\beta_{0}=-\gamma_{1},\qquad\beta_{1}=-\frac{1}{4}h^{(\bar{Q})}+\frac{1}{8}\gamma_{0}^{2}$
$\displaystyle\qquad\Rightarrow\quad h^{(Q)}=-2\gamma_{0}\gamma_{1}.$

####  5.1.2 The array of Poisson brackets

We constructed the 3 canonical representations (5.4), (5.5c) and (5.6c),
respectively on spaces $\bf q$, $\bf Q$ and $\bf\bar{Q}$. The canonical
brackets are then extended to include the parameters as Casimirs:

$\displaystyle{\mathcal{P}}^{(q)}_{0}={\mathcal{P}}^{(Q)}_{1}={\mathcal{P}}^{(\bar{Q})}_{2}=\left(\begin{matrix}\phantom{-}0&1&0&0\\\
-1&0&0&0\\\ \phantom{-}0&0&0&0\\\ \phantom{-}0&0&0&0\end{matrix}\right).$

Following the same procedure as Section 4.1.1, we construct the other 6
Poisson brackets. From $P_{0}^{(q)}$, we obtain

$\displaystyle{\mathcal{P}}^{(Q)}_{0}$
$\displaystyle=\frac{1}{4\beta_{0}}\left(\begin{matrix}\phantom{-}0&\phantom{-}2P_{1}&a_{13}&0\\\
-2P_{1}&\phantom{-}0&a_{23}&0\\\ -a_{13}&-a_{23}&0&0\\\
\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$
$\displaystyle{\mathcal{P}}^{(\bar{Q})}_{0}$
$\displaystyle=\frac{1}{4\gamma_{1}^{2}}\left(\begin{matrix}0&\gamma_{0}\bar{Q}_{1}\bar{P}_{1}-2\gamma_{1}\bar{Q}_{1}-\gamma_{0}^{2}&-\gamma_{0}b_{13}&\gamma_{1}b_{13}\\\
\gamma_{0}^{2}+2\gamma_{1}\bar{Q}_{1}-\gamma_{0}\bar{Q}_{1}\bar{P}_{1}&0&-\gamma_{0}b_{23}&\gamma_{1}b_{23}\\\
\gamma_{0}b_{13}&\gamma_{0}b_{23}&0&0\\\
-\gamma_{1}b_{13}&-\gamma_{1}b_{23}&0&0\end{matrix}\right),$

where $(a_{13},a_{23},0,0)^{\rm T}={\mathcal{P}}^{(Q)}_{1}\nabla_{Q}h^{(Q)}$
and $(b_{13},b_{23},0,0)^{\rm
T}={\mathcal{P}}^{(\bar{Q})}_{2}\nabla_{\bar{Q}}h^{(\bar{Q})}$.

From $P_{1}^{(Q)}$, we obtain

$\displaystyle{\mathcal{P}}^{(q)}_{1}=\left(\begin{matrix}\phantom{-}0&\phantom{-}q_{1}&a_{13}&0\\\
-q_{1}&\phantom{-}0&a_{23}&0\\\ -a_{13}&-a_{23}&0&0\\\
\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),\qquad{\mathcal{P}}^{(\bar{Q})}_{1}=\frac{1}{2\gamma_{1}}\left(\begin{matrix}\phantom{-}0&\bar{Q}_{1}\bar{P}_{1}-\gamma_{0}&b_{13}&0\\\
\gamma_{0}-\bar{Q}_{1}\bar{P}_{1}&\phantom{-}0&b_{23}&0\\\
-b_{13}&-b_{23}&0&0\\\ \phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$

where $(a_{13},a_{23},0,0)^{\rm T}={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}$
and $(b_{13},b_{23},0,0)^{\rm
T}=-{\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}$.

From $P_{2}^{(\bar{Q})}$, we obtain

$\displaystyle{\mathcal{P}}^{(Q)}_{2}=\left(\begin{matrix}\phantom{-}0&-Q_{1}&0&a_{14}\\\\[2.84526pt]
\phantom{-}Q_{1}&\phantom{-}0&0&a_{24}\\\ \phantom{-}0&\phantom{-}0&0&0\\\
-a_{14}&-a_{24}&0&0\end{matrix}\right),\qquad{\mathcal{P}}^{(q)}_{2}=\left(\begin{matrix}\phantom{-}0&-q_{1}^{2}&0&b_{14}\\\\[5.69054pt]
\phantom{-}q_{1}^{2}&\phantom{-}0&0&b_{24}\\\\[2.84526pt]
\phantom{-}0&\phantom{-}0&0&0\\\ -b_{14}&-b_{24}&0&0\end{matrix}\right),$

where $(a_{14},a_{24},0,0)^{\rm
T}=-\frac{1}{4}{\mathcal{P}}^{(Q)}_{1}\nabla_{Q}h^{(Q)}$ and
$(b_{14},b_{24},0,0)^{\rm
T}=\frac{1}{2}{\mathcal{P}}^{(q)}_{0}\nabla_{q}h^{(q)}$.

Each of these has 2 Casimirs and the $t_{1}=t_{h}$ flow in each space has a
tri-Hamiltonian representation.

In the $\bf q$ space, we have

$\displaystyle{\bf
q}_{t_{h}}={\mathcal{P}}_{2}^{(q)}\nabla_{q}(2\alpha_{1})={\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{0}={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},$
$\displaystyle{\mathcal{P}}_{2}^{(q)}\nabla_{q}\alpha_{0}={\mathcal{P}}_{2}^{(q)}\nabla_{q}h^{(q)}={\mathcal{P}}_{1}^{(q)}\nabla_{q}h^{(q)}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}=0.$

In the $\bf Q$ space, we have

$\displaystyle{\bf
Q}_{t_{h}}={\mathcal{P}}_{2}^{(Q)}\nabla_{Q}(-4\beta_{1})={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\left(2\beta_{0}^{2}\right),$
$\displaystyle{\mathcal{P}}_{2}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{2}^{(Q)}\nabla_{Q}\beta_{0}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta_{0}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta_{1}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\beta_{1}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0.$

In the $\bf\bar{Q}$ space, we have

$\displaystyle{\bf\bar{Q}}_{t_{h}}={\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\left(h^{(\bar{Q})}-\frac{1}{2}\gamma_{0}^{2}\right)={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}(-2\gamma_{0}\gamma_{1})={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(2\gamma_{1}^{2}\right),$
$\displaystyle{\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}={\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\left(h^{(\bar{Q})}-\frac{1}{2}\gamma_{0}^{2}\right)$
$\displaystyle\hphantom{{\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}}{}={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(h^{(\bar{Q})}-\frac{1}{2}\gamma_{0}^{2}\right)={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}(\gamma_{0}\gamma_{1})=0.$

######  Remark 5.1 (relation of (5.5c) to (4.6)).

The canonical transformation

$\displaystyle Q_{1}=\frac{1}{2}\tilde{Q}_{1}^{2},\qquad
P_{1}=\frac{\tilde{Q}_{1}\tilde{P}_{1}+\sqrt{-2\beta}}{\tilde{Q}_{1}^{2}},\qquad\text{with}\quad\alpha=-4\beta_{1},\quad\beta=-8\beta_{0}^{2},$

gives $h^{(Q)}=\frac{1}{2}h^{(\tilde{Q})}$, where
$h^{(\tilde{Q})}=\frac{1}{2}\tilde{P}_{1}^{2}-\frac{1}{8}\tilde{Q}_{1}^{6}+\alpha\tilde{Q}_{1}^{2}+\frac{\beta}{\tilde{Q}_{1}^{2}}$,
which is just (4.6) with relabelled variables.

This transformation is real when $\beta<0$.

Under this transformation, ${\mathcal{P}}_{i}^{(Q)}$ are transformed to
${\mathcal{P}}_{i}^{(\tilde{Q})}$, with
${\mathcal{P}}_{1}^{(\tilde{Q})}={\mathcal{P}}_{1}^{(Q)}$ and

$\displaystyle{\mathcal{P}}^{(\tilde{Q})}_{0}=2\left(\begin{matrix}\phantom{-}0&\phantom{-}\frac{1}{\tilde{Q}_{1}^{2}}&a_{13}&0\\\
-\frac{1}{\tilde{Q}_{1}^{2}}&\phantom{-}0&a_{23}&0\\\ -a_{13}&-a_{23}&0&0\\\
\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),\qquad{\mathcal{P}}^{(\tilde{Q})}_{2}=\frac{1}{2}\left(\begin{matrix}\phantom{-}0&-\tilde{Q}_{1}^{2}&0&b_{14}\\\\[5.69054pt]
\phantom{-}\tilde{Q}_{1}^{2}&\phantom{-}0&0&b_{24}\\\\[2.84526pt]
\phantom{-}0&\phantom{-}0&0&0\\\ -b_{14}&-b_{24}&0&0\end{matrix}\right),$

where $(a_{13},a_{23},0,0)^{\rm
T}=-{\mathcal{P}}^{(\tilde{Q})}_{1}\nabla_{\tilde{Q}}h^{(\tilde{Q})}$ and
$(b_{14},b_{24},0,0)^{\rm
T}={\mathcal{P}}^{(\tilde{Q})}_{1}\nabla_{\tilde{Q}}h^{(\tilde{Q})}$, thus
rendering the flow of $h^{(\tilde{Q})}$ as tri-Hamiltonian

$\displaystyle\tilde{\bf
Q}_{t_{h}}={\mathcal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}(2\alpha)={\mathcal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}h^{(\tilde{Q})}={\mathcal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(-\frac{1}{2}\beta\right).$

This system also has the Lax matrix (4.13a), with $N=1$.

###  5.2 The $\boldsymbol{t_{2}}$ flow

We now derive the explicit formulae for the case $n=2$.

The $u$ space, with (5.1): We previously derived this as (4.7b), giving

$\displaystyle
h^{(q)}=5q_{2}p_{1}^{2}-4p_{1}p_{2}-\frac{q_{2}}{64}\big{(}4q_{1}+q_{2}^{2}\big{)}\big{(}12q_{1}+7q_{2}^{2}\big{)}+\alpha_{0}q_{2}+\alpha_{1}\big{(}4q_{1}+q_{2}^{2}\big{)},$
(5.7a) where $u_{0}=q_{1}$, $u_{1}=q_{2}$, $u_{0x}=2(5q_{2}p_{1}-2p_{2})$,
$u_{1x}=-4p_{1}$. In (4.7c), we also gave the first integral $\displaystyle
f^{(q)}=\frac{1}{256}\big{(}4q_{1}+3q_{2}^{2}\big{)}\big{(}64(\alpha_{0}-2\alpha_{1}q_{2})+64p_{1}^{2}-16q_{1}^{2}+5q_{2}^{4}\big{)}-2(p_{2}-q_{2}p_{1})^{2}.$
(5.7b)

The $w$ space, with (5.2): Using $2H_{3}^{w}$, we have

$\displaystyle{\mathcal{L}}_{2}^{w}=\frac{1}{2}w_{0x}^{2}-\frac{1}{8}w_{1x}^{2}-\frac{3}{4}w_{1}^{2}w_{0x}+\frac{1}{2}w_{0}^{4}-\frac{3}{4}w_{0}^{2}w_{1}^{2}+\frac{5}{32}w_{1}^{4}-\beta_{0}w_{0}-\beta_{1}w_{1}.$
(5.8a) Defining $Q_{1}=w_{0}$, $Q_{2}=w_{1}$,
$P_{1}=w_{0x}-\frac{3}{4}w_{1}^{2}$, $P_{2}=-\frac{1}{4}w_{1x}$, we obtain the
Hamiltonian $\displaystyle
h^{(Q)}=\frac{1}{2}P_{1}^{2}-2P_{2}^{2}+\frac{3}{4}Q_{2}^{2}P_{1}-\frac{1}{2}Q_{1}^{4}+\frac{3}{4}Q_{1}^{2}Q_{2}^{2}+\frac{1}{8}Q_{2}^{4}+\beta_{0}Q_{1}+\beta_{1}Q_{2}.$
(5.8b)

######  Remark 5.2 (first integral).

Rather than calculating fluxes of the modified PDEs, we derive the first
integral $f^{(Q)}$ (also $f^{(\bar{Q})}$ below) directly through the Miura
maps in the stationary coordinates, later in our calculations.

The $\boldsymbol{v}$ space, with (5.3): Using $2H_{2}^{v}$, we have

$\displaystyle{\mathcal{L}}_{2}^{v}=-\frac{1}{4}v_{1x}^{3}-\frac{3}{2}v_{0}v_{1}v_{1x}^{2}+v_{0x}v_{1x}-3v_{0}^{2}v_{1}^{2}v_{1x}+2v_{0}^{3}v_{1}\big{(}1-v_{1}^{2}\big{)}-\gamma_{0}v_{0}-\gamma_{1}v_{1}.$
(5.9a) Defining $\bar{Q}_{1}=v_{0}$, $\bar{Q}_{2}=v_{1}$,
$\bar{P}_{1}=v_{1x}$,
$\bar{P}_{2}=-\frac{3}{4}v_{1x}^{2}-3v_{0}v_{1}v_{1x}+v_{0x}-3v_{0}^{2}v_{1}^{2}$,
we obtain the Hamiltonian $\displaystyle
h^{(\bar{Q})}=\frac{1}{4}\bar{P}_{1}^{3}+\bar{P}_{1}\bar{P}_{2}+\frac{3}{2}\bar{Q}_{1}\bar{Q}_{2}\bar{P}_{1}^{2}+3\bar{Q}_{1}^{2}\bar{Q}_{2}^{2}\bar{P}_{1}+2\bar{Q}_{1}^{3}\bar{Q}_{2}\big{(}\bar{Q}_{2}^{2}-1\big{)}+\gamma_{0}\bar{Q}_{1}+\gamma_{1}\bar{Q}_{2}.$
(5.9b)

####  5.2.1 The Miura maps in these coordinates

We consider the two steps induced by ${\bf w}\mapsto{\bf u}$ and ${\bf
v}\mapsto{\bf w}$. We again extend each space to include the parameters as
dynamical variables, which we define as ${\bf
q}=(q_{1},q_{2},p_{1},p_{2},\alpha_{0},\alpha_{1})$, ${\bf
Q}=(Q_{1},Q_{2},P_{1},P_{2},\beta_{0},\beta_{1})$,
${\bf\bar{Q}}=\big{(}\bar{Q}_{1},\bar{Q}_{2},\bar{P}_{1},\bar{P}_{2},\gamma_{0},\gamma_{1}\big{)}$.

The relation of $\bf q$ to $\bf Q$: found by using the Miura map and the
definitions of $\bf q$ and $\bf Q$. Formulae for $\alpha_{0}$, $\alpha_{1}$
are derived from the equations of motion implied by $h^{(q)}$. In summary

$\displaystyle
q_{1}=-P_{1}-Q_{1}^{2}-\frac{3}{4}Q_{2}^{2},\qquad\\!q_{2}=Q_{2},\qquad\\!p_{1}=P_{2},\qquad\\!p_{2}=\frac{1}{2}Q_{1}\big{(}P_{1}+Q_{1}^{2}\big{)}+Q_{2}P_{2}-\frac{1}{4}\beta_{0},$
$\displaystyle\alpha_{1}=\frac{1}{4}\beta_{1},\qquad\alpha_{0}=\frac{1}{2}h^{(Q)},\qquad
h^{(q)}=-\frac{1}{2}f^{(Q)},\quad f^{(q)}=-\frac{1}{8}\beta_{0}^{2},$

where

$\displaystyle
f^{(Q)}=\frac{1}{4}\big{(}P_{1}+Q_{1}^{2}\big{)}\big{(}8\beta_{1}+16Q_{1}P_{2}+4Q_{2}P_{1}+\big{(}8Q_{1}^{2}+Q_{2}^{2}\big{)}Q_{2}\big{)}-\beta_{0}(2P_{2}+Q_{1}Q_{2}),$

satisfying $\big{\\{}h^{(Q)},f^{(Q)}\big{\\}}=0$.

The relation of $\bf Q$ to $\bf\bar{Q}$: found by using the Miura map and the
definitions of $\bf Q$ and $\bf\bar{Q}$. Formulae for $\beta_{0}$, $\beta_{1}$
are derived from the equations of motion implied by $h^{(Q)}$. In summary

$\displaystyle Q_{1}=\\!\bar{Q}_{1},\qquad
Q_{2}=-\bar{P}_{1}-2\bar{Q}_{1}\bar{Q}_{2},$ $\displaystyle
P_{1}=\\!\bar{P}_{2},\qquad
P_{2}=\frac{1}{2}\big{(}\bar{Q}_{1}\bar{P}_{1}+\bar{Q}_{2}\bar{P}_{2}\big{)}+\frac{3}{2}\bar{Q}_{1}^{2}\bar{Q}_{2}-\frac{1}{4}\gamma_{0},$
$\displaystyle\beta_{0}=\gamma_{1},\qquad\beta_{1}=\frac{1}{2}h^{(\bar{Q})},\qquad
h^{(Q)}=\frac{1}{4}f^{(\bar{Q})}-\frac{1}{8}\gamma_{0}^{2},\qquad
f^{(Q)}=\frac{1}{2}\gamma_{0}\gamma_{1},$

where

$\displaystyle
f^{(\bar{Q})}=\big{(}\bar{P}_{2}+\bar{Q}_{1}^{2}\big{)}\big{(}\bar{P}_{1}\big{(}\bar{P}_{1}+4\bar{Q}_{1}\bar{Q}_{2}\big{)}-2\big{(}\bar{P}_{2}-\bar{Q}_{1}^{2}\big{)}\big{(}\bar{Q}_{2}^{2}-1\big{)}\big{)}+2\gamma_{0}\bar{Q}_{2}\big{(}\bar{P}_{2}+\bar{Q}_{1}^{2}\big{)}$
$\displaystyle\hphantom{f^{(\bar{Q})}=}-2\gamma_{1}\big{(}\bar{Q}_{2}\bar{P}_{1}+2\bar{Q}_{1}\big{(}\bar{Q}_{2}^{2}-1\big{)}\big{)},$

satisfying $\big{\\{}h^{(\bar{Q})},f^{(\bar{Q})}\big{\\}}=0$.

####  5.2.2 The array of Poisson brackets

We constructed the 3 canonical representations (5.7a), (5.8b) and (5.9b),
respectively on spaces $\bf q$, $\bf Q$ and $\bf\bar{Q}$. The canonical
brackets are then extended to include the parameters $\alpha_{i}$,
$\beta_{i}$, $\gamma_{i}$ as Casimirs:

$\displaystyle{\mathcal{P}}^{(q)}_{0}={\mathcal{P}}^{(Q)}_{1}={\mathcal{P}}^{(\bar{Q})}_{2}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&1&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&1&0&0\\\ -1&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&-1&0&0&0&0\\\ \phantom{-}0&\phantom{-}0&0&0&0&0\\\
\phantom{-}0&\phantom{-}0&0&0&0&0\end{matrix}\right).$

Following the same procedure as Section 5.1.2, we construct the other 6
Poisson brackets.

From $P_{0}^{(q)}$, we obtain

$\displaystyle{\mathcal{P}}^{(Q)}_{0}=\frac{1}{4\beta_{0}}\\!\left(\\!\begin{matrix}\phantom{-}0&-16Q_{1}&8(2P_{2}+Q_{1}Q_{2})&4P_{1}\\!+\\!8Q_{1}^{2}\\!+\\!3Q_{2}^{2}&a_{15}&0\\\
16Q_{1}&\phantom{-}0&-32Q_{1}^{2}&\phantom{-}0&a_{25}&0\\\
-8(2P_{2}+Q_{1}Q_{2})&\phantom{-}32Q_{1}^{2}&\phantom{-}0&\phantom{-}a_{34}&a_{35}&0\\\
-\big{(}4P_{1}\\!+\\!8Q_{1}^{2}\\!+\\!3Q_{2}^{2}\big{)}&\phantom{-}0&-a_{34}&\phantom{-}0&a_{45}&0\\\
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\end{matrix}\\!\right),$
$\displaystyle{\mathcal{P}}^{(\bar{Q})}_{0}=\frac{1}{\gamma_{1}^{2}}\left(\\!\begin{matrix}\phantom{-}0&\phantom{-}b_{12}&\phantom{-}b_{13}&\gamma_{1}\big{(}2\bar{Q}_{2}\big{(}\bar{P}_{2}+\bar{Q}_{1}^{2}\big{)}-\gamma_{0}\big{)}&b_{15}&b_{16}\\\
-b_{12}&\phantom{-}0&\phantom{-}b_{23}&\phantom{-}b_{24}&b_{25}&b_{26}\\\
-b_{13}&-b_{23}&\phantom{-}0&\phantom{-}b_{34}&b_{35}&b_{36}\\\
\gamma_{1}\big{(}\gamma_{0}-2\bar{Q}_{2}\big{(}\bar{P}_{2}+\bar{Q}_{1}^{2}\big{)}\big{)}&-b_{24}&-b_{34}&\phantom{-}0&b_{45}&b_{46}\\\
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0&0\\\
-b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0\end{matrix}\\!\right),$

where
$a_{34}=-2\big{(}2\beta_{0}+Q_{1}\big{(}4P_{1}+8Q_{1}^{2}+3Q_{2}^{2}\big{)}\big{)}$,
$(a_{15},a_{25},a_{35},a_{45},0,0)^{\rm
T}=8{\mathcal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}$ and

$\displaystyle
b_{12}=\gamma_{0}\big{(}\bar{P}_{1}+2\bar{Q}_{1}\bar{Q}_{2}\big{)}-2\gamma_{1}\bar{Q}_{2}^{2},\qquad
b_{13}=2\bar{Q}_{1}\big{(}2\gamma_{1}\big{(}\bar{Q}_{2}^{2}+1\big{)}-\gamma_{0}\big{(}{\bar{P}}_{1}+2{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}\big{)},$
$\displaystyle
b_{23}=2\big{(}\gamma_{0}({\bar{P}}_{1}+2{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}\bar{Q}_{2}-2\gamma_{1}\bar{Q}_{2}\big{(}\bar{Q}_{2}^{2}-1\big{)}-2\gamma_{0}\bar{Q}_{1}\big{)},$
$\displaystyle
b_{24}=2\gamma_{0}\big{(}{\bar{P}}_{1}+{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}\bar{Q}_{1}-2\gamma_{1}\big{(}{\bar{P}}_{1}+4{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}\bar{Q}_{2}-2\gamma_{0}\bar{Q}_{2}\bar{P}_{2}+4\gamma_{1}\bar{Q}_{1}+\gamma_{0}^{2},$
$\displaystyle
b_{34}=2\big{(}2\gamma_{1}\bar{Q}_{1}\bar{Q}_{2}\big{(}{\bar{P}}_{1}+3{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}-2\gamma_{0}\big{(}{\bar{P}}_{1}+{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}\bar{Q}_{1}^{2}+\big{(}\gamma_{0}\bar{Q}_{1}-\gamma_{1}\bar{Q}_{2}\big{)}\big{(}2\bar{Q}_{2}\bar{P}_{2}-\gamma_{0}\big{)}\big{)},$
$\displaystyle(b_{15},b_{25},b_{35},b_{45},0,0)^{\rm
T}={\mathcal{P}}^{(\bar{Q})}_{2}\nabla_{\bar{Q}}\big{(}\gamma_{0}f^{({\bar{Q}})}-4\gamma_{1}h^{({\bar{Q}})}\big{)},$
$\displaystyle(b_{16},b_{26},b_{36},b_{46},0,0)^{\rm
T}=-{\mathcal{P}}^{(\bar{Q})}_{2}\nabla_{\bar{Q}}\big{(}\gamma_{1}f^{({\bar{Q}})}\big{)}.$

From $P_{1}^{(Q)}$, we obtain
$\big{(}\frac{\partial\bf{q}}{\partial\bf{Q}}\big{)}{\mathcal{P}}^{(Q)}_{1}\big{(}\frac{\partial\bf{q}}{\partial\bf{Q}}\big{)}^{\rm
T}=-\frac{1}{8}{\mathcal{P}}^{(q)}_{1}$, of (4.11b), and

$\displaystyle{\mathcal{P}}^{(\bar{Q})}_{1}=\frac{1}{\gamma_{1}}\left(\begin{matrix}\phantom{-}0&-\big{(}\bar{P}_{1}+2\bar{Q}_{1}\bar{Q}_{2}\big{)}&2\bar{Q}_{1}\big{(}\bar{P}_{1}+2\bar{Q}_{1}\bar{Q}_{2}\big{)}&\phantom{-}\gamma_{1}&a_{15}&0\\\
\bar{P}_{1}+2\bar{Q}_{1}\bar{Q}_{2}&\phantom{-}0&\phantom{-}a_{23}&\phantom{-}a_{24}&a_{25}&0\\\
-2\bar{Q}_{1}\big{(}\bar{P}_{1}+2\bar{Q}_{1}\bar{Q}_{2}\big{)}&-a_{23}&\phantom{-}0&\phantom{-}a_{34}&a_{35}&0\\\
-\gamma_{1}&-a_{24}&-a_{34}&\phantom{-}0&a_{45}&0\\\
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\end{matrix}\right),$

where

$\displaystyle
a_{23}=2\big{(}2\bar{Q}_{1}-\bar{Q}_{2}\bar{P}_{1}-2\bar{Q}_{1}\bar{Q}_{2}^{2}\big{)},\qquad
a_{24}=2\big{(}\bar{Q}_{2}\bar{P}_{2}-\bar{Q}_{1}\bar{P}_{1}-\bar{Q}_{1}^{2}\bar{Q}_{2}\big{)}-\gamma_{0},$
$\displaystyle
a_{34}=2\big{(}2\bar{Q}_{1}^{2}\big{(}{\bar{P}}_{1}+{\bar{Q}}_{1}{\bar{Q}}_{2}\big{)}-2\bar{Q}_{1}\bar{Q}_{2}\bar{P}_{2}+\gamma_{0}\bar{Q}_{1}-\gamma_{1}\bar{Q}_{2}\big{)},$
$\displaystyle(a_{15},a_{25},a_{35},a_{45},0,0)^{\rm
T}=-{\mathcal{P}}^{(\bar{Q})}_{2}\nabla_{\bar{Q}}f^{({\bar{Q}})}.$

From $P_{2}^{(\bar{Q})}$, we obtain $\big{(}\frac{\partial{\bf
Q}}{\partial{\bf\bar{Q}}}\big{)}{\mathcal{P}}_{2}^{(\bar{Q})}\big{(}\frac{\partial{\bf
Q}}{\partial{\bf\bar{Q}}}\big{)}^{\rm T}=\frac{1}{2}{\mathcal{P}}_{2}^{(Q)}$
and $\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}{\mathcal{P}}_{2}^{(Q)}\big{(}\frac{\partial{\bf q}}{\partial{\bf
Q}}\big{)}^{\rm T}=\frac{1}{4}{\mathcal{P}}_{2}^{(q)}$:

$\displaystyle{\mathcal{P}}^{(Q)}_{2}=\left(\begin{matrix}\phantom{-}0&-2&\phantom{-}0&\phantom{-}Q_{1}&0&a_{16}\\\\[2.84526pt]
\phantom{-}2&\phantom{-}0&-4Q_{1}&-Q_{2}&0&a_{26}\\\
\phantom{-}0&\phantom{-}4Q_{1}&\phantom{-}0&-(P_{1}+3Q_{1}^{2})&0&a_{36}\\\
-Q_{1}&\phantom{-}Q_{2}&P_{1}+3Q_{1}^{2}&\phantom{-}0&0&a_{46}\\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\\\
-a_{16}&-a_{26}&-a_{36}&-a_{46}&0&0\end{matrix}\right),$
$\displaystyle{\mathcal{P}}^{(q)}_{2}=\left(\begin{matrix}\phantom{-}0&\phantom{-}0&3q_{2}^{2}-4q_{1}&\frac{1}{2}q_{2}(4q_{1}+15q_{2}^{2})&b_{15}&b_{16}\\\
\phantom{-}0&\phantom{-}0&-4q_{2}&-4q_{1}-7q_{2}^{2}&b_{25}&b_{26}\\\
4q_{1}-3q_{2}^{2}&\phantom{-}4q_{2}&\phantom{-}0&\phantom{-}4q_{2}p_{1}&b_{35}&b_{36}\\\
-\frac{1}{2}q_{2}(4q_{1}+15q_{2}^{2})&4q_{1}+7q_{2}^{2}&-4q_{2}p_{1}&\phantom{-}0&b_{45}&b_{46}\\\
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0&0\\\
-b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0\end{matrix}\right),$

where

$\displaystyle(a_{16},\dots,0)^{\rm
T}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)},\qquad(b_{15},\dots,0)^{\rm
T}=4{\mathcal{P}}_{0}^{(q)}\nabla_{q}f^{(q)},$
$\displaystyle(b_{16},\dots,0)^{\rm
T}={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}.$

Each of these has 2 Casimirs and the flows in each space have tri-Hamiltonian
representations.

In the $\bf q$ space, we have

$\displaystyle{\bf
q}_{t_{h}}={\mathcal{P}}_{2}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{4}\alpha_{0}\right)={\mathcal{P}}_{0}^{(q)}\nabla_{q}h^{(q)},$
$\displaystyle{\bf
q}_{t_{f}}={\mathcal{P}}_{2}^{(q)}\nabla_{q}\left(\frac{1}{4}\alpha_{0}\right)={\mathcal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{8}h^{(q)}\right)={\mathcal{P}}_{0}^{(q)}\nabla_{q}f^{(q)},$
$\displaystyle{\mathcal{P}}_{2}^{(q)}\nabla_{q}h^{(q)}={\mathcal{P}}_{2}^{(q)}\nabla_{q}f^{(q)}={\mathcal{P}}_{1}^{(q)}\nabla_{q}f^{(q)}={\mathcal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}={\mathcal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}=0.$

In the $\bf Q$ space, we have

$\displaystyle{\bf
Q}_{t_{h}}={\mathcal{P}}_{2}^{(Q)}\nabla_{Q}\beta_{1}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\left(-\frac{1}{2}f^{(Q)}\right),$
$\displaystyle{\bf
Q}_{t_{f}}={\mathcal{P}}_{2}^{(Q)}\nabla_{Q}(-h^{(Q)})={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\left(\frac{1}{4}\beta_{0}^{2}\right),$
$\displaystyle{\mathcal{P}}_{2}^{(Q)}\nabla_{Q}f^{(Q)}={\mathcal{P}}_{2}^{(Q)}\nabla_{Q}\beta_{0}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta_{0}={\mathcal{P}}_{1}^{(Q)}\nabla_{Q}\beta_{1}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}\beta_{1}={\mathcal{P}}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0.$

In the $\bf\bar{Q}$ space, we have

$\displaystyle{\bf\bar{Q}}_{t_{h}}={\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\left(\frac{1}{4}f^{(\bar{Q})}-\frac{1}{8}\gamma_{0}^{2}\right)={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(-\frac{1}{4}\gamma_{0}\gamma_{1}\right),$
$\displaystyle{\bf\bar{Q}}_{t_{f}}={\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}f^{(\bar{Q})}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\left(-\gamma_{0}\gamma_{1}\right)={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(-\frac{1}{2}\gamma_{1}^{2}\right),$
$\displaystyle{\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}={\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\mathcal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}$
$\displaystyle\hphantom{{\mathcal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}}{}={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}={\mathcal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(f^{(\bar{Q})}-\frac{1}{2}\gamma_{0}^{2}\right)=0.$

######  Remark 5.3 (relation of (5.8b) to (4.10)).

The canonical transformation

$\displaystyle
Q_{1}=\frac{2^{\frac{2}{3}}\big{(}\sqrt{-2\beta}-\tilde{Q}_{2}\tilde{P}_{2}\big{)}}{\tilde{Q}_{2}^{2}},\qquad
Q_{2}=2^{\frac{2}{3}}\tilde{Q}_{1},$ $\displaystyle
P_{1}=\frac{\tilde{Q}_{2}^{2}}{2^{\frac{5}{3}}}-2^{\frac{4}{3}}\frac{\big{(}\sqrt{-2\beta}-\tilde{Q}_{2}\tilde{P}_{2}\big{)}^{2}}{\tilde{Q}_{2}^{4}},\qquad
P_{2}=\\!\frac{\tilde{P}_{1}}{2^{\frac{2}{3}}},$ (5.10a) with
$\alpha=\beta_{1}$, $\beta=-\frac{1}{2}\beta_{0}^{2}$, gives
$h^{(Q)}=-2^{\frac{2}{3}}h^{(\tilde{Q})}$,
$f^{(Q)}=-2^{\frac{1}{3}}f^{(\tilde{Q})}$, where $\displaystyle
h^{(\tilde{Q})}=\frac{1}{2}\big{(}\tilde{P}_{1}^{2}+\tilde{P}_{2}^{2}\big{)}-\alpha\tilde{Q}_{1}-\frac{1}{32}\big{(}16\tilde{Q}_{1}^{4}+12\tilde{Q}_{1}^{2}\tilde{Q}_{2}^{2}+\tilde{Q}_{2}^{4}\big{)}+\frac{\beta}{\tilde{Q}_{2}^{2}},$
(5.10b) $\displaystyle
f^{(\tilde{Q})}=\tilde{P}_{1}(\tilde{Q}_{1}\tilde{P}_{1}+\tilde{Q}_{2}\tilde{P}_{2})-2\tilde{Q}_{1}h^{(\tilde{Q})}-\frac{\alpha}{2}\big{(}4\tilde{Q}_{1}^{2}+\tilde{Q}_{2}^{2}\big{)}$
$\displaystyle\hphantom{f^{(\tilde{Q})}=}{}-\frac{1}{16}\tilde{Q}_{1}\big{(}4\tilde{Q}_{1}^{2}+\tilde{Q}_{2}^{2}\big{)}\big{(}4\tilde{Q}_{1}^{2}+3\tilde{Q}_{2}^{2}\big{)},$
(5.10c) which are just (4.10) and (4.8b) (for $N=2$), with relabelled
variables $\tilde{\bf
Q}=\big{(}\tilde{Q}_{i},\tilde{P}_{i},\beta,\alpha\big{)}$.

This transformation is real when $\beta<0$.

Under this transformation, ${\mathcal{P}}_{i}^{(Q)}$ are transformed to
${\mathcal{P}}_{i}^{(\tilde{Q})}$, with
${\mathcal{P}}_{1}^{(\tilde{Q})}={\mathcal{P}}_{1}^{(Q)}$ and (up to overall
numerical factors)

$\displaystyle{\mathcal{P}}^{(\tilde{Q})}_{0}=\begin{pmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{\tilde{Q}_{2}}&a_{15}&0\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}\frac{1}{\tilde{Q}_{2}}&-\frac{2\tilde{Q}_{1}}{\tilde{Q}_{2}^{2}}&a_{25}&0\\\\[5.69054pt]
\phantom{-}0&-\frac{1}{\tilde{Q}_{2}}&\phantom{-}0&\phantom{-}\frac{\tilde{P}_{2}}{\tilde{Q}_{2}^{2}}&a_{35}&0\\\\[5.69054pt]
-\frac{1}{\tilde{Q}_{2}}&\phantom{-}\frac{2\tilde{Q}_{1}}{\tilde{Q}_{2}^{2}}&-\frac{\tilde{P}_{2}}{\tilde{Q}_{2}^{2}}&\phantom{-}0&a_{45}&0\\\\[5.69054pt]
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\end{pmatrix},$
$\displaystyle{\mathcal{P}}^{(\tilde{Q})}_{2}=\begin{pmatrix}\phantom{-}0&\phantom{-}0&\phantom{-}2\tilde{Q}_{1}&\phantom{-}\tilde{Q}_{2}&0&b_{16}\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}\tilde{Q}_{2}&\phantom{-}0&0&b_{26}\\\\[5.69054pt]
-2\tilde{Q}_{1}&-\tilde{Q}_{2}&\phantom{-}0&\phantom{-}\tilde{P}_{2}&0&b_{36}\\\\[5.69054pt]
-\tilde{Q}_{2}&\phantom{-}0&-\tilde{P}_{2}&\phantom{-}0&0&b_{46}\\\\[5.69054pt]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&0&0\\\\[5.69054pt]
-b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0\end{pmatrix},$ (5.10d)

where $(a_{15},a_{25},a_{35},a_{45},0,0)^{\rm
T}=-{\mathcal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}f^{(\tilde{Q})}$ and
$(b_{16},b_{26},b_{36},b_{46},0,0)^{\rm
T}=2{\mathcal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}h^{(\tilde{Q})}$, thus
rendering the flow of (5.10b) as tri-Hamiltonian:

$\displaystyle\tilde{\bf
Q}_{t_{h}}={\mathcal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(\frac{1}{2}\alpha\right)={\mathcal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}h^{(\tilde{Q})}={\mathcal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}f^{(\tilde{Q})}.$
(5.10e)

This system also has the Lax matrix (4.14a), with $N=2$.

##  6 Generalisations: coupling with the Calogero–Moser model

Several of our Lax pairs can be generalised (when $N=3$) to incorporate an
arbitrary function in the Hamiltonian, which allows us to relate the system to
the rational Calogero–Moser model, following an approach described in [13].

In Section 6.1, we present generalisations of the Garnier and Hénon–Heiles
systems, as well as the Hamiltonian (4.8a), with quartic potential, but
others, such as (4.4c), can be similarly generalised. For each case we give
three functions, $h^{(B)}$, $f^{(B)}$ and $h_{23}^{(B)}$, which are in
involution, along with Lax matrices.

The connection to the Calogero–Moser model is explained in Section 6.2.

###  6.1 Generalisations in $\boldsymbol{(Q_{i},P_{i})}$ coordinates

####  6.1.1 Generalised Garnier system

In Section 3.1, we derived two versions of the multicomponent Garnier system,
given by (3.5b) and (3.5c). For brevity, we just give the generalisation of
the second of these:

$\displaystyle
h^{(B)}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\big{)}+k\big{(}Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}^{2}+\frac{\beta}{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
$\displaystyle
f^{(B)}=(Q_{1}P_{2}-Q_{2}P_{1})^{2}+(Q_{2}P_{3}-Q_{3}P_{2})^{2}+(Q_{3}P_{1}-Q_{1}P_{3})^{2}+2\frac{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}{Q_{2}^{2}}B\\!\left(\frac{Q_{3}}{Q_{2}}\right),$
$\displaystyle
h_{23}^{(B)}=(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2\frac{\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$

where $B\big{(}\frac{Q_{3}}{Q_{2}}\big{)}$ is an arbitrary function. When
$k=-\frac{1}{4}$ and $B\big{(}\frac{Q_{3}}{Q_{2}}\big{)}=0$, these reduce to
(3.5c) and elements of the rotation algebra.

The Lax matrix (3.18a), with $N=3$ and $\sum\frac{2\beta_{i}}{Q_{i}^{2}}$
replaced by $\frac{2\beta}{{\bf Q}^{2}}$, can similarly be generalised:

$\displaystyle
L^{(1)}=\begin{pmatrix}Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3}&2\lambda-
Q_{1}^{2}-Q_{2}^{2}-Q_{3}^{2}\\\ a_{21}&-(Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3})\\\
\end{pmatrix}$ $\displaystyle\qquad\Rightarrow\quad
z^{2}+16k\lambda^{3}-4h^{(B)}\lambda+f^{(B)}+2\beta=0,$

where

$a_{21}=-4k\lambda\big{(}2\lambda+Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}+P_{1}^{2}+P_{2}^{2}+P_{3}^{2}+\frac{2\beta}{Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}}+\frac{2}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right).$

The Lax equations (3.18b) are unchanged, but $U$ is deformed:

$\displaystyle U=\left(\begin{matrix}0&1\\\
-4k\big{(}\lambda+Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}&0\end{matrix}\right)\quad\Rightarrow\quad\left\\{\begin{matrix}L^{(1)}_{x}=\big{\\{}L^{(1)},h^{(B)}\big{\\}}=\big{[}U,L^{(1)}\big{]},\\\\[2.84526pt]
\big{\\{}L^{(1)},f^{(B)}\big{\\}}=\big{\\{}L^{(1)},h_{23}^{(B)}\big{\\}}=0.\end{matrix}\right.$

####  6.1.2 Generalised Hénon–Heiles system

The generalised Hénon–Heiles potential of (3.12b) was considered in [13,
Section 3.5], for the case $N=3$, with some additional terms:

$\displaystyle
h^{(B)}=\frac{1}{2}\big{(}P_{1}^{2}\\!+\\!P_{2}^{2}\\!+\\!P_{3}^{2}\big{)}\\!+\\!\frac{\omega^{2}}{2}\big{(}4Q_{1}^{2}\\!+\\!Q_{2}^{2}\\!+\\!Q_{3}^{2}\big{)}\\!+\\!kQ_{1}\big{(}2Q_{1}^{2}\\!+\\!Q_{2}^{2}\\!+\\!Q_{3}^{2}\big{)}\\!+\\!\frac{1}{Q_{2}^{2}}B\\!\left(\frac{Q_{3}}{Q_{2}}\right),$
(6.1a) $\displaystyle
f^{(B)}=P_{1}(Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3})-2Q_{1}h^{(B)}+2\omega^{2}Q_{1}\big{(}2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}$
$\displaystyle\hphantom{f^{(B)}=}{}+\frac{k}{4}\big{(}16Q_{1}^{4}+12Q_{1}^{2}\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}+\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}^{2}\big{)},$
(6.1b) $\displaystyle
h_{23}^{(B)}=(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2\frac{\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(6.1c)

where $B\left(\frac{Q_{3}}{Q_{2}}\right)$ is an arbitrary function.

When $B(z)=\beta_{2}+\frac{\beta_{3}}{z^{2}}$, $k=\frac{1}{2}$, $\omega=0$,
then $h^{(B)}$ reduces to $h^{(Q)}$ of (3.12b). We also have that $f^{(B)}$
reduces to $f^{(Q)}$ and $h_{23}^{(B)}$ to $h_{23}+2(\beta_{2}+\beta_{3})$.

The Lax matrix (3.19a), with $N=3$, can similarly be generalised:

$\displaystyle L^{(2)}=\left(\begin{matrix}Q_{2}P_{2}+Q_{3}P_{3}-4\lambda
P_{1}&16\lambda^{2}+8\lambda Q_{1}-Q_{2}^{2}-Q_{3}^{2}\\\ a_{21}&4\lambda
P_{1}-Q_{2}P_{2}-Q_{3}P_{3}\end{matrix}\right),$ (6.2a) where $\displaystyle
a_{21}=2k\big{(}16\lambda^{3}-8Q_{1}\lambda^{2}\\!+\lambda\big{(}4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}\big{)}+8\omega^{2}\lambda(Q_{1}-2\lambda)+P_{2}^{2}+P_{3}^{2}+\frac{2}{Q_{2}^{2}}B\\!\left(\frac{Q_{3}}{Q_{2}}\right)\\!,$
and the characteristic equation is $\displaystyle
z^{2}=512k\lambda^{5}-256\omega^{2}\lambda^{4}+32h^{(B)}\lambda^{2}-8f^{(B)}\lambda-
h_{23}^{(B)}.$ (6.2b) The Lax equations are just the same as (3.19c):
$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(B)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(B)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$
$\displaystyle\big{\\{}L^{(2)},h_{23}^{(B)}\big{\\}}=0,$ (6.2c) but with
deformed versions of $U$ and $L^{(1)}$: $\displaystyle
U=\left(\begin{matrix}0&1\\\ 2k(\lambda-
Q_{1})-\omega^{2}&0\end{matrix}\right),\qquad
L^{(1)}=\left(\begin{matrix}P_{1}&-2(2\lambda+Q_{1})\\\
b_{21}&-P_{1}\end{matrix}\right),$ (6.2d) where
$b_{21}=k\big{(}{-}8\lambda^{2}+4\lambda
Q_{1}-\big{(}2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}\big{)}+2\omega^{2}(2\lambda-
Q_{1}).$

####  6.1.3 Generalisation of the Hamiltonian (4.8a)

The potential of (4.8a) was labelled $U_{4}$ in [13, Section 3.3], for the
case $N=3$. With some additional terms, and writing
$\beta=\frac{1}{2}\omega^{2}$, we have

$\displaystyle
h^{(B)}=\frac{1}{2}\big{(}P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\big{)}-\alpha
Q_{1}+\frac{\omega^{2}}{2}\big{(}4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}$
$\displaystyle\hphantom{h^{(B)}=}{}+k\big{(}16Q_{1}^{4}+12Q_{1}^{2}\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}+\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}^{2}\big{)}+\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(6.3a) $\displaystyle
f^{(B)}=P_{1}(Q_{1}P_{1}+Q_{2}P_{2}+Q_{3}P_{3})-2Q_{1}h^{(B)}-\frac{\alpha}{2}\big{(}4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}$
$\displaystyle\hphantom{f^{(B)}=}{}+2\omega^{2}Q_{1}\big{(}2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}+2kQ_{1}\big{(}4Q_{1}^{2}+3Q_{2}^{2}+3Q_{3}^{2}\big{)}\big{(}4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)},$
(6.3b) $\displaystyle
h_{23}^{(B)}=(Q_{2}P_{3}-Q_{3}P_{2})^{2}+2\frac{\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right),$
(6.3c)

where $B\big{(}\frac{Q_{3}}{Q_{2}}\big{)}$ is an arbitrary function.

When $B(z)=\beta_{2}+\frac{\beta_{3}}{z^{2}}$, $k=-\frac{1}{32}$, $\omega=0$,
then $h^{(B)}$ reduces to $h^{(Q)}$ of (4.8a). We also have that $f^{(B)}$
reduces to $f^{(Q)}$ and $h_{23}^{(B)}$ to $h_{23}+2(\beta_{2}+\beta_{3})$.

The Lax matrix (4.14a), with $N=3$, is similarly generalised:

$\displaystyle L^{(2)}=\left(\begin{matrix}Q_{2}P_{2}+Q_{3}P_{3}-4\lambda
P_{1}&16\lambda^{2}+8\lambda Q_{1}-\big{(}Q_{2}^{2}+Q_{3}^{2}\big{)}\\\
a_{21}&4\lambda P_{1}-(Q_{2}P_{2}+Q_{3}P_{3})\end{matrix}\right),$

where

$\displaystyle
a_{21}=P_{2}^{2}+P_{3}^{2}+\frac{2}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)+4\lambda\big{(}2\omega^{2}(Q_{1}-2\lambda)-\alpha\big{)}$
$\displaystyle\hphantom{a_{21}=}{}-32k\lambda\big{(}16\lambda^{3}-8\lambda^{2}Q_{1}+\lambda\big{(}4Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}-Q_{1}\big{(}2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}\big{)}.$

The characteristic equation of $L^{(2)}$ is

$\displaystyle
z^{2}+8192k\lambda^{6}+256\omega^{2}\lambda^{4}+64\alpha\lambda^{3}-32h^{(B)}\lambda^{2}+8f^{(B)}\lambda+h_{23}^{(B)}=0.$

The Lax equations generated by $h^{(B)}$, $f^{(B)}$ and $h_{23}^{(B)}$ are the
same as (4.14c):

$\displaystyle
L^{(2)}_{x}=\big{\\{}L^{(2)},h^{(B)}\big{\\}}=\big{[}U,L^{(2)}\big{]},\qquad
L^{(2)}_{t_{1}}=\big{\\{}L^{(2)},f^{(B)}\big{\\}}=\big{[}L^{(1)},L^{(2)}\big{]},$
$\displaystyle\big{\\{}L^{(2)},h_{23}^{(B)}\big{\\}}=0,$

where $L^{(1)}$ and $U$ are deformations of (4.14d):

$\displaystyle L^{(1)}=\begin{pmatrix}P_{1}&-4\lambda-2Q_{1}\\\
b_{21}&-P_{1}\\\ \end{pmatrix},\qquad U=\begin{pmatrix}0&1\\\
-32k\big{(}\lambda^{2}-\lambda
Q_{1}+\frac{1}{8}\big{(}6Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}\big{)}-\omega^{2}&0\\\
\end{pmatrix},$

with

$\displaystyle b_{21}=\alpha+2\omega^{2}(2\lambda-Q_{1})$
$\displaystyle\hphantom{b_{21}=}{}-32k\big{(}\tfrac{1}{2}Q_{1}\big{(}Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}-\tfrac{1}{2}\lambda\big{(}2Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}\big{)}+2Q_{1}\lambda^{2}-4\lambda^{3}\big{)}.$

######  Remark 6.1.

The cases (6.1a) and (6.3a) belong to a family of Hamiltonians which are
separable in generalised parabolic coordinates [13].

###  6.2 Transformation to Calogero–Moser coordinates

The Hamiltonian $h^{(B)}$, with just the $B$ term (for example, (6.1a) with
$k=\omega=0$) is known [9, 15] to have 4 integrals for general $B$, with an
additional integral for the particular choice
$B(z)=\frac{9g^{2}(1+z^{2})^{2}}{2(1-3z^{2})^{2}}$. This particular choice,
with the (orthogonal) canonical transformation, generated by

$\displaystyle
S=\frac{1}{\sqrt{3}}(q_{1}+q_{2}+q_{3})P_{1}+\frac{1}{\sqrt{2}}(q_{1}-q_{2})P_{2}+\frac{1}{\sqrt{6}}(q_{1}+q_{2}-2q_{3})P_{3},$

gives the rational Calogero–Moser potential:

$\displaystyle\frac{1}{Q_{2}^{2}}B\left(\frac{Q_{3}}{Q_{2}}\right)=g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right).$

In [13], we gave a large class of additional potentials which could be added
to $B$ and thus coupled with the rational Calogero–Moser system, but had no
Lax representations. In this paper, we see that any system written in Section
6.1 gives rise to a coupling of the rational Calogero–Moser system with a Lax
representation. To illustrate this, we just present the Hénon–Heiles case.

####  6.2.1 Calogero–Moser system, coupled with the Hénon–Heiles potential

Since the formulae are more complicated in the Calogero–Moser coordinates, we
introduce three functions which give some simplifications:

$\displaystyle\tau=q_{1}+q_{2}+q_{3},\qquad\rho=q_{1}^{2}+q_{2}^{2}+q_{3}^{2},\qquad\delta=(q_{1}-q_{2})^{2}+(q_{1}-q_{3})^{2}+(q_{2}-q_{3})^{2}.$

Under this canonical transformation, the functions $h^{(B)}$, $f^{(B)}$,
$h_{23}^{(B)}$ (of (6.1)) take the form

$\displaystyle
h^{{\rm(CM)}}=\frac{1}{2}\big{(}p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\big{)}+g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right)$
$\displaystyle\hphantom{h^{{\rm(CM)}}=}{}+\frac{k}{3\sqrt{3}}\tau(6\rho-\delta)+\frac{\omega^{2}}{2}(4\rho-\delta),$
$\displaystyle
f^{{\rm(CM)}}=\frac{1}{\sqrt{3}}(p_{1}+p_{2}+p_{3})(q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3})-\frac{2\tau}{\sqrt{3}}h^{{\rm(CM)}}$
$\displaystyle\hphantom{f^{{\rm(CM)}}=}{}+\frac{2}{3\sqrt{3}}\omega^{2}\tau(6\rho-\delta)+\frac{k}{36}\big{(}144\rho^{2}+5\delta^{2}-60\rho\delta\big{)},$
$\displaystyle
h_{23}^{\rm(CM)}=\frac{1}{3}((q_{2}-q_{3})p_{1}+(q_{3}-q_{1})p_{2}+(q_{1}-q_{2})p_{3})^{2}$
$\displaystyle\hphantom{h_{23}^{\rm(CM)}=}{}+\frac{2}{3}g^{2}\delta\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right).$

The Lax matrices of Section 6.1.2 take the form

$\displaystyle L^{(2)}=\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&-a_{11}\end{pmatrix},\qquad U=\begin{pmatrix}0&1\\\
2k\left(\lambda-\frac{1}{\sqrt{3}}\tau\right)-\omega^{2}&0\end{pmatrix},$
$\displaystyle
L^{(1)}=\frac{1}{\sqrt{3}}\begin{pmatrix}p_{1}+p_{2}+p_{3}&-4\sqrt{3}\lambda-2\tau\\\
\sqrt{3}b_{21}&-(p_{1}+p_{2}+p_{3})\end{pmatrix},$

where

$\displaystyle
a_{11}=q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3}-\frac{1}{3}\big{(}\tau+4\sqrt{3}\lambda\big{)}(p_{1}+p_{2}+p_{3}),\qquad
a_{12}=16\lambda^{2}+\frac{8}{\sqrt{3}}\tau\lambda-\frac{1}{3}\delta,$
$\displaystyle
a_{21}=32k\left(\lambda^{3}-\frac{1}{2\sqrt{3}}\tau\lambda^{2}+\frac{1}{16}\lambda(4\rho-\delta)\right)+\frac{8\omega^{2}\lambda}{\sqrt{3}}\big{(}\tau-2\sqrt{3}\lambda\big{)}$
$\displaystyle\hphantom{a_{21}=}{}+\frac{2}{3}\big{(}p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}\big{)}$
$\displaystyle\hphantom{a_{21}=}{}+2g^{2}\left(\frac{1}{(q_{1}-q_{2})^{2}}+\frac{1}{(q_{1}-q_{3})^{2}}+\frac{1}{(q_{2}-q_{3})^{2}}\right),$
$\displaystyle
b_{21}=\frac{2\omega^{2}}{\sqrt{3}}\big{(}2\sqrt{3}\lambda-\tau\big{)}-k\left(8\lambda^{2}-\frac{4\tau}{\sqrt{3}}\lambda+\frac{1}{3}(6\rho-\delta)\right).$

The characteristic equation of $L^{(2)}$ and the Lax equations are just the
same as (6.2b) and (6.2c), but written in terms of these coordinates. Setting
$k=0$, the system reduces to the resonant harmonic oscillator case presented
in [13, Section 4.2].

This Lax matrix is certainly not as elegant as the usual Calogero–Moser one
[17], but it does include the additional potentials. Furthermore, the
Calogero–Moser system is known to be superintegrable [21] and $f^{(CM)}$,
$h_{23}^{(CM)}$ are related to his “additional” integrals, rather than those
generated by the usual Lax matrix.

##  7 Conclusions

In this paper we have reconsidered the relationship of integrable nonlinear
evolution equations and their stationary flows, which define finite-
dimensional Hamiltonian systems (also integrable).

Multicomponent squared eigenfunction expansions gave us Hamiltonians with
higher degrees of freedom, such as (3.5c), (3.12b), (4.4c), (4.8a) and their
rotationally symmetric versions. Restricting the dimension allowed us to build
Poisson maps, giving us bi-Hamiltonian representations, by comparing the
definitions of the corresponding canonical variables. The space is extended to
include some arbitrary parameters as dynamical variables and the Poisson maps
are non-canonical. For the DWW hierarchy, this was extended to a tri-
Hamiltonian representation in Section 5. This used the DWW Miura maps, so (for
each time-evolution) gave 3 tri-Hamiltonian systems. In particular, we showed
that the well known Hamiltonian (5.10b), with quartic potential (see [14,
19]), is tri-Hamiltonian.

For each of these systems, we presented a $2\times 2$ Lax representation,
derived from the zero-curvature representation of the corresponding coupled
KdV equation. For the case of 3 degrees of freedom, these Lax representations
are further generalised in Section 6 to connect with the results of [13],
where we studied the class of Hamiltonians separable by generalised parabolic
coordinates. A large subclass of these have the arbitrary function
$B\big{(}\frac{Q_{3}}{Q_{2}}\big{)}$, introduced in Section 6. This gave a way
of deriving Lax pairs and first integrals for a coupling of the rational
Calogero–Moser model to many other potential functions, such as the
Hénon–Heiles potential, given in Section 6.2.1.

From their derivation, the Lax matrices of this paper inherit a polynomial
$\lambda$-dependence, with complicated coefficients. Clearly a larger matrix
with linear $\lambda$-dependence (such as a deformation of the usual Lax
matrix of [17]) would be preferable. Currently these generalisations are
restricted to 3 degrees of freedom, related to the results of [13]. For higher
degrees of freedom, we would like generalisations with simple reductions to
the 3 degrees of freedom case. These are quite varied and not necessarily
related to the Calogero–Moser model.

In this paper we only considered the cases $M=1$ and $M=2$ in (2.1a), mainly
presenting details of the first two nontrivial stationary flows. The squared
eigenfunction representations (3.3b) and (4.2), corresponding to $h^{(Q)}$ of
Sections 3 and 4, is the most interesting.

The $t_{1}$ flow has the general structure

$\displaystyle
h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)+U\left(\sum_{i=1}^{N}Q_{i}^{2}\right),$
(7.1a) and has a universal set of integrals, given by (3.6a). For $N=3$, this
is the first case in Table 2 of [11]. For general $N$, this class of
Hamiltonian (on a curved space background) has been analysed in [5], where the
same set of universal integrals was derived. The particular form of the
function $U$ in (7.1a), derived in our construction, depends upon the value of
$M$ in (2.1a), but universally possesses a Lax representation.

The $t_{2}$ flow has the general structure

$\displaystyle
h^{(Q)}=\frac{1}{2}P_{1}^{2}+\frac{1}{2}\sum_{i=2}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)+U\left(Q_{1},\sum_{i=2}^{N}Q_{i}^{2}\right),$
(7.1b)

which also has the universal set of integrals (3.6a), but for $2\leq i<j\leq
N$, as well as the integral $f^{(Q)}$. For $N=2$ we just have the integral
$f^{(Q)}$, but we see that the potential belongs to the class separable in
parabolic coordinates (see [18, equation (2.2.41)]). From our construction, we
have two additional features: $M+1$ compatible Poisson brackets (for $N=2$)
and a Lax representation (for all $N$).

The $t_{2}$ flow is particularly interesting and leads to a number of
important questions: Can we extend the multi-Poisson formulation beyond $N=2$?
Even for $N=2$, can the multi-Poisson formulation be extended to the entire
class of potentials separable in parabolic coordinates? Can the Lax pair be
similarly extended? Some insight into these questions may be obtained by
considering the case of (2.1a) with $M=3$.

### Acknowledgements

This work was supported by the National Natural Science Foundation of China
(grant no. 11871396).

## References

  * [1] Antonowicz M., Fordy A.P., Coupled KdV equations with multi-Hamiltonian structures, Phys. D 28 (1987), 345–357.
  * [2] Antonowicz M., Fordy A.P., Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (1989), 465–486.
  * [3] Antonowicz M., Fordy A.P., Wojciechowski S., Integrable stationary flows: Miura maps and bi-Hamiltonian structures, Phys. Lett. A 124 (1987), 143–150.
  * [4] Baker S., Enolskii V.Z., Fordy A.P., Integrable quartic potentials and coupled KdV equations, Phys. Lett. A 201 (1995), 167–174, arXiv:hep-th/9504087.
  * [5] Ballesteros A., Enciso A., Herranz F.J., Ragnisco O., Superintegrability on $N$-dimensional curved spaces: central potentials, centrifugal terms and monopoles, Ann. Physics 324 (2009), 1219–1233, arXiv:0812.1882.
  * [6] Bogojavlenskii O.I., Novikov S.P., The connection between the Hamiltonian formalisms of stationary and nonstationary problems, Funct. Anal. Appl. 10 (1976), 8–11.
  * [7] Calogero F., Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419–436.
  * [8] Cariglia M., Hidden symmetries of dynamics in classical and quantum physics, Rev. Modern Phys. 86 (2014), 1283–1333, arXiv:math-ph/1411.1262.
  * [9] Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  * [10] Dorizzi B., Grammaticos B., Hietarinta J., Ramani A., Schwarz F., New integrable three-dimensional quartic potentials, Phys. Lett. A 116 (1986), 432–436.
  * [11] Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666–5676.
  * [12] Fordy A.P., The Hénon–Heiles system revisited, Phys. D 52 (1991), 204–210.
  * [13] Fordy A.P., Huang Q., Integrable and superintegrable extensions of the rational Calogero–Moser model in three dimensions, J. Phys. A 55 (2022), 225203, 36 pages, arXiv:2111.15659.
  * [14] Hietarinta J., A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phys. Lett. A 96 (1983), 273–278.
  * [15] Horwood J.T., Higher order first integrals in classical mechanics, J. Math. Phys. 48 (2007), 102902, 18 pages.
  * [16] Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), 423001, 97 pages, arXiv:math-ph/1309.2694.
  * [17] Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197–220.
  * [18] Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Vol. I, Birkhäuser, Basel, 1990.
  * [19] Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture revisited, Phys. Rev. Lett. 49 (1982), 1539–1541.
  * [20] Reyman A.G., Semenov-Tian-Shansky M.A., Compatible Poisson structures for Lax equations: an $r$-matrix approach, Phys. Lett. A 130 (1988), 456–460.
  * [21] Wojciechowski S., Superintegrability of the Calogero–Moser system, Phys. Lett. A 95 (1983), 279–281.
  * [22] Wojciechowski S., Integrability of one particle in a perturbed central quartic potential, Phys. Scripta 31 (1985), 433–438.