Separability and Symmetry Operators for Painlevé Metrics

Separability and Symmetry Operators for Painlevé
Metrics and their Conformal Deformations

Thierry DAUDÉ †, Niky KAMRAN ‡ and Francois NICOLEAU §

T. Daudé, N. Kamran and F. Nicoleau

† Département de Mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise,
† 95302 Cergy-Pontoise, France<EMAIL_ADDRESS>
‡ Department of Mathematics and Statistics, McGill University,
‡ Montreal, QC, H3A 2K6, Canada<EMAIL_ADDRESS>
§ Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629,
§ 2 Rue de la Houssinière BP 92208, F-44322 Nantes Cedex 03, France
<EMAIL_ADDRESS>
Received March 27, 2019, in final form September 05, 2019; Published online
September 16, 2019

Painlevé metrics are a class of Riemannian metrics which generalize the well-
known separable metrics of Stäckel to the case in which the additive
separation of variables for the Hamilton–Jacobi equation is achieved in terms
of groups of independent variables rather than the complete orthogonal
separation into ordinary differential equations which characterizes the
Stäckel case. Painlevé metrics in dimension $n$ thus admit in general only
$r<n$ linearly independent Poisson-commuting quadratic first integrals of the
geodesic flow, where $r$ denotes the number of groups of variables. Our goal
in this paper is to carry out for Painlevé metrics the generalization of the
analysis, which has been extensively performed in the Stäckel case, of the
relation between separation of variables for the Hamilton–Jacobi and Helmholtz
equations, and of the connections between quadratic first integrals of the
geodesic flow and symmetry operators for the Laplace–Beltrami operator. We
thus obtain the generalization for Painlevé metrics of the Robertson
separability conditions for the Helmholtz equation which are familiar from the
Stäckel case, and a formulation thereof in terms of the vanishing of the off-
block diagonal components of the Ricci tensor, which generalizes the one
obtained by Eisenhart for Stäckel metrics. We also show that when the
generalized Robertson conditions are satisfied, there exist $r<n$ linearly
independent second-order differential operators which commute with the
Laplace–Beltrami operator and which are mutually commuting. These operators
admit the block-separable solutions of the Helmholtz equation as formal
eigenfunctions, with the separation constants as eigenvalues. Finally, we
study conformal deformations which are compatible with the separation into
blocks of variables of the Helmholtz equation for Painlevé metrics, leading to
solutions which are $R$-separable in blocks. The paper concludes with a set of
open questions and perspectives.

Painlevé metrics; Killing tensors; Helmholtz equation; $R$-separability;
symmetry operators; Robertson conditions

53B21; 70H20; 81Q80

##  1 Introduction and statement of results

Painlevé metrics [37, 38] are a class of Riemannian metrics that provide a
broad generalization of the well-known separable metrics of Stäckel [18, 41]
to the case in which the Hamilton–Jacobi equation for the geodesic flow can be
additively separated into partial differential equations depending on groups
of independent variables rather ordinary differential equations resulting from
a complete orthogonal separation. In particular, while Stäckel metrics in
dimension $n$ admit $n$ linearly independent Poisson-commuting quadratic first
integrals of the geodesic flow, Painlevé metrics in dimension $n$ will admit
only $r<n$ such integrals in general, where $r$ denotes the number of groups
of variables.

Our goal in this paper is to carry out the extension to Painlevé metrics of
the well-known results [1, 2, 3, 18, 25, 26, 27] which relate in the Stäckel
case the additive separation of variables for the Hamilton–Jacobi equation to
the multiplicative separation of variables for the Helmholtz equation, and the
existence of quadratic first integrals of the geodesic flow to that of
symmetry operators for the Laplace–Beltrami operator. We shall thus obtain the
generalization to Painlevé metrics of the Robertson separability conditions
[39] for the Helmholtz equation for Stäckel metrics, and a formulation of
these generalized Robertson conditions in terms of the vanishing of the off-
block diagonal components of the Ricci tensor, thereby extending the classical
result proved by Eisenhart [18] for Stäckel metrics. We shall also show that
when the generalized Robertson conditions are satisfied, there exist $r<n$
linearly independent second-order differential operators which commute between
themselves and with the Laplace–Beltrami operator. These operators will be
shown to admit the block-separable solutions of the Helmholtz equation as
formal eigenfunctions, with the separation constants arising from the
separation into groups variables as eigenvalues. Finally, we shall also
discuss conformal deformations of Painlevé metrics satisfying a further
generalization of the Robertson conditions, which is compatible with the
separation into blocks of variables of the Helmholtz equation, leading to
solutions which are $R$-separable in blocks.

Before describing our results in further detail, we should remark that
independently of the interest of Painlevé metrics from the point of view of
separation of variables, a key motivation for our study lies in the goal of
constructing geometric models of manifolds with boundary endowed with Painlevé
metrics, with the goal of investigating the anisotropic Calderón problem in
this class of geometries. Recall that the anisotropic Calderón problem
consists in recovering the metric of a Riemannian manifold with boundary from
the knowledge of the Dirichlet-to-Neumann map defined by the Laplace–Beltrami
operator. The anisotropic Calderón problem is at the center of a great amount
of current research activity. We refer to [12, 15, 22, 28, 29, 32, 33, 40, 46,
47] and the references therein for surveys of recent results on this problem.
We have recently investigated the anisotropic Calderón problem at fixed energy
for geometric models consisting of classes of Stäckel manifolds with boundary,
where the separation of variables for the Helmholtz equation allows the
decomposition of the Dirichlet-to-Neumann map onto a basis of joint
eigenfunctions of the symmetry operators resulting from the complete
separation of variables, enabling us to obtain a series of uniqueness and non-
uniqueness results for the Calderón problem, with no a-priori assumptions of
analyticity, or on the existence of isometries [11, 12, 13, 14, 16, 21]. In
the case of Painlevé metrics, the separation of the Helmholtz into groups of
variables and the concomitant families of commuting symmetry operators
admitted by these metrics will serve as an effective starting point for the
investigation of the anisotropic Calderón problem in this more general
setting.

In order to put the results of the present paper in context, we first briefly
recall some well-known definitions and results pertaining to Stäckel metrics
and their separability properties. Throughout the paper, we shall assume for
simplicity that the manifolds, metrics and maps being considered are smooth,
although many of the results that we quote or obtain can be shown to hold
under weaker differentiability properties. Recall [2, 18, 25, 41] that a
_Stäckel metric_ on an $n$-dimensional manifold $M$ is a Riemannian metric $g$
for which there exist local coordinates $\big{(}x^{1},\dots,x^{n}\big{)}$ in
which the metric has the expression

$\displaystyle{\rm d}s^{2}=g_{ij}{\rm d}x^{i}{\rm d}x^{j}=\frac{\det
S}{s^{11}}\big{(}{\rm d}x^{1}\big{)}^{2}+\cdots+\frac{\det
S}{s^{n1}}\big{(}{\rm d}x^{n}\big{)}^{2},$ (1.1)

where $S$ is a _Stäckel matrix_ , that is a non-singular $n\times n$ matrix
$S=(s_{ij})$ of the form

$\displaystyle
S=\left(\begin{matrix}s_{11}\big{(}x^{1}\big{)}&\dots&s_{1n}\big{(}x^{1}\big{)}\\\
\vdots&&\vdots\\\
s_{n1}\big{(}x^{n}\big{)}&\dots&s_{nn}\big{(}x^{n}\big{)}\end{matrix}\right),$
(1.2)

and $s^{ij}$ denotes the cofactor of the component $s_{ij}$ of the matrix $S$.
Stäckel matrices thus have the property that for each $1\leq i\leq n$, their
$i$-th row depends only on the $i$-th local coordinate $x^{i}$, and that the
cofactor $s^{ij}$ is independent of the $i$-th local coordinate $x^{i}$.
Furthermore, the diagonal components of the Stäckel metric (1.1) are given by
the inverses of the entries of the first row of the inverse Stäckel matrix
$A=S^{-1}$.

The importance of Stäckel metrics stems from the fact that they constitute the
most general class of Riemannian metrics admitting orthogonal coordinates for
which the Hamilton–Jacobi equation

$\displaystyle g^{ij}\partial_{i}W\partial_{j}W=E,$ (1.3)

for the geodesic flow of $(M,g)$, where $E$ denotes a positive real constant,
admits a complete integral obtained by additive separation of variables into
ordinary differential equations. It is useful at this stage to recall that a
complete integral of (1.3) is defined as a parametrized family of solutions

$\displaystyle W=W\big{(}x^{1},\dots,x^{n};a_{1},\dots,a_{n}\big{)},\qquad
a_{1}:=E,$ (1.4)

defined over the domain $U\subset M$ of the local coordinates
$\big{(}x^{1},\dots,x^{n}\big{)}$ and depending smoothly on $n$ parameters
$(a_{1},\dots,a_{n})$ defined on an open subset $A\subset{\mathbb{R}}^{n}$,
such that the rank condition

$\displaystyle\det\left(\frac{\partial^{2}W}{\partial x^{i}\partial
a_{j}}\right)\neq 0,$

is satisfied throughout the open set $U\times A$.

It is easily verified that the Hamilton–Jacobi equation (1.3) will admit an
additively separable complete integral $W\big{(}x^{1},\dots,x^{n};a_{1},\dots
a_{n}\big{)}$ of the form

$W=W_{1}\big{(}x^{1};a_{1},\dots,a_{n}\big{)}+\cdots+W_{n}\big{(}x^{n};a_{1},\dots,a_{n}\big{)},$

if and only if the summands $W_{\alpha}$ satisfy the set of separated ordinary
differential equations given by

$\left(\frac{{\rm d}W_{i}}{{\rm
d}x^{i}}\right)^{2}=\sum_{j=1}^{n}s_{ij}\big{(}x^{i}\big{)}a_{j}.$

The $n$ parameters $(a_{1},\dots,a_{n})$ appearing in the expression of the
additively separable complete integral (1.4) thus correspond to the separation
constants arising from the complete separation of variables of the
Hamilton–Jacobi equation into ordinary differential equations. One of the key
consequences of this complete separation of variables property is that the
geodesic flow of an $n$-dimensional Stäckel metric admits a linearly
independent set of $n-1$ mutually Poisson-commuting quadratic first integrals,
given by

$\displaystyle
K_{(l)}=K_{(l)}^{ij}p_{i}p_{j}=\sum_{j=1}^{n}a_{lj}p_{j}^{2},\qquad 2\leq
l\leq n$ (1.5)

with

$\big{\\{}K_{(l)},K_{(m)}\big{\\}}=0,\qquad\text{for}\quad 1\leq l,m\leq n,$

where $A=(a_{ij})$ denotes as above the inverse of the Stäckel matrix $S$
given by (1.2). Note that with the notations of (1.5), we have $K_{(1)}=H$,
where

$H=g^{ij}p_{i}p_{j},$

denotes the Hamiltonian for the geodesic flow.

A question closely related to the additive separation of the Hamilton–Jacobi
equation is that of the complete multiplicative separation of the Helmholtz
equation

$\displaystyle-\Delta_{g}u=\lambda u,$ (1.6)

where

$\Delta_{g}=\frac{1}{\sqrt{|g|}}\partial_{i}\Big{(}\sqrt{|g|}g^{ij}\partial_{j}\Big{)},\qquad|g|=\det(g_{ij}),$

denotes the Laplace–Beltrami operator on $(M,g)$ and $\lambda$ denotes a non
vanishing real constant, into ordinary differential equations. By complete
multiplicative separation, we mean, following [2], the existence of a
parametrized family of solutions $u$ defined on a domain $U\subset M$ with
local coordinates $\big{(}x^{1},\dots,x^{n}\big{)}$ of the form

$u\big{(}x^{1},\dots,x^{n};a_{1},\dots,a_{2n}\big{)}=\prod_{i=1}^{n}u_{i}\big{(}x^{1},\dots,x^{n};a_{1},\dots,a_{2n}\big{)},$

depending smoothly on $2n$ parameters $(a_{1},\dots,a_{2n})$ defined on open
subset $A\subset{\mathbb{R}}^{2n}$, satisfying the rank condition

$\displaystyle\det\left(\begin{matrix}\dfrac{\partial v_{1}}{\partial
a_{1}}&\dots&\dfrac{\partial v_{1}}{\partial a_{2n}}\\\ \vdots&&\vdots\\\
\dfrac{\partial v_{n}}{\partial a_{1}}&\dots&\dfrac{\partial v_{n}}{\partial
a_{2n}}\vspace{1mm}\\\ \dfrac{\partial w_{1}}{\partial
a_{1}}&\dots&\dfrac{\partial w_{1}}{\partial a_{2n}}\\\ \vdots&&\vdots\\\
\dfrac{\partial w_{n}}{\partial a_{1}}&\dots&\dfrac{\partial w_{n}}{\partial
a_{2n}}\end{matrix}\right)\neq 0,$

at every point of $U\times A$, where

$v_{i}=\frac{u_{i}^{\prime}}{u_{i}}\qquad
w_{i}=\frac{u_{i}^{\prime\prime}}{u_{i}}.$

This separation requires that additional conditions, known as the _Robertson
conditions_ , and given by

$\partial_{i}\Gamma_{j}=0,\qquad 1\leq i\neq j\leq n,$

where

$\Gamma_{i}=-\partial_{i}\left[\log\frac{(\det
S)^{\frac{n}{2}-1}s^{i1}}{\big{(}s^{11}\cdots
s^{n1}\big{)}^{\frac{1}{2}}}\right],$

be satisfied. We refer to [2, 3, 18, 25, 26, 39] for detailed proofs of the
fact that the Robertson conditions are necessary and sufficient for complete
multiplicative separation of the Helmholtz equation. The Robertson conditions
were interpreted by Eisenhart [18] in terms of the vanishing of the off-
diagonal components of the Ricci tensor of the underlying Stäckel metric, that
is

$\displaystyle R_{ij}=0\qquad\text{for}\qquad 1\leq i\neq j\leq n.$ (1.7)

When the Robertson conditions are satisfied, the Poisson-commuting quadratic
first integrals of the geodesic flow give rise to $n-1$ linearly independent
second-order differential operators which commute with the Laplace–Beltrami
operator and also commute pairwise. Rewriting the quadratic first integrals
$K_{(l)}$ defined by (1.5) in the form

$K_{(l)}=K_{(l)}^{ij}p_{i}p_{j},$

these commuting operators, denoted by $\Delta_{K_{(l)}}$, are of the form

$\Delta_{K_{(l)}}=\nabla_{i}\big{(}K_{(l)}^{ij}\nabla_{j}\big{)},\qquad 2\leq
l\leq n,$

where $\nabla_{i}$ denotes the Levi-Civita connection on $(M,g)$. These
operators, which are often referred to as _symmetry operators_ , admit the
separable solutions of the Helmholtz equation as (formal) eigenfunctions. We
will not give any further details on symmetry operators at this stage, nor
shall we say anything about the proofs of the results we have just recalled
since we shall shortly state and prove generalizations of these to the case of
Painlevé metrics, which admit all Stäckel metrics as a special case.

We conclude these preliminaries by remarking that the above setting may be
expanded significantly by considering conformal deformations of Stäckel
metrics which are compatible with the complete separation of the Helmholtz
equation into ordinary differential equations, thus giving rise to the more
general notion of _$R$ -separability_ for the Helmholtz equation. Again, we
shall not give any additional details on these topics at this stage since
conformal deformations and $R$-separability will be studied in the remainder
of this paper in the more general setting of Painlevé metrics. We refer to [2,
3, 4, 8, 9, 25, 26, 27] for lucid accounts of the key results on the
separability and $R$-separability properties of Stäckel metrics, their
connection to Killing tensors, quadratic first integrals of the geodesic flow
and symmetry operators for the Laplace–Beltrami operator. We also refer to [1,
36] for recent surveys on separability on Riemannian manifolds and to [7] for
a penetrating analysis of the relations between quadratic first integrals of
the geodesic flow, symmetry operators and conserved currents, in the general
setting of Riemannian or pseudo-Riemannian manifolds.

With these preliminaries at hand, we are now ready to introduce the class of
Painlevé metrics [37, 38]. As stated above, Painlevé metrics arise as a
natural generalization of Stäckel metrics to the case in which one no longer
seeks complete separation of the Hamilton–Jacobi equation into ordinary
differential equations, but rather separation into partial differential
equations involving groups of variables. The separable coordinates admitted by
Painlevé metrics are thus generally _not orthogonal_ , although they are
orthogonal with respect to groups of variables. Let us recall that our goal in
this paper is to carry out for Painlevé metrics the analogue of the
separability and $R$-separability analyses of the Helmholtz equation which has
been extensively worked out for Stäckel metrics in [2, 3, 8, 25, 26, 27], and
to show that the separability into groups of variables gives rise to vector
spaces of mutually commuting symmetry operators for the Laplace–Beltrami
operator, the dimension of which is determined by the number of groups of
variables. In particular, we will generalize to the case of Painlevé metrics
the Robertson conditions and the characterization thereof in terms of the
Ricci tensor. We now proceed to define the class of Painlevé metrics along
lines similar to the ones used above for Stäckel metrics.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold and let ${\bf
x}=\big{(}x^{1},\dots,x^{n}\big{)}$ denote a set of local coordinates on $M$.
We shall consider partitions of ${\bf x}$ into $r\geq 2$ groups of local
coordinates,

${\bf x}=\big{(}{{\bf x}^{1}},\dots,{{\bf x}^{r}}\big{)},$

where

${\bf{x}}^{\alpha}=\big{(}x^{i_{\alpha}}\big{)},\qquad 1_{\alpha}\leq
i_{\alpha}\leq
l_{\alpha},\qquad\text{and}\qquad\sum_{\alpha=1}^{r}||l_{\alpha}||=n,$

where $||l_{\alpha}||$ denotes the number of local coordinates in the group of
label $\alpha$. Latin indices $1\leq i,j,\ldots\leq n$ will be used to label
the local coordinates on $M$, greek indices $\alpha,\beta,\dots$ to label the
$r$ groups of local coordinates, and hybrid indices $i_{\alpha}$,
$1_{\alpha}\leq i_{\alpha}\leq l_{\alpha}$ to denote the local coordinates
within the group $\bf{x^{\alpha}}$. Unless there is an ambiguity in the
notation being used, in which case we will write out the summation signs
explicitly, we shall apply the summation convention with the above range of
indices. A generalized Stäckel matrix is a non-singular $r\times r$ matrix-
valued function $S$ on $M$ of the form

$\displaystyle S=\left(\begin{matrix}s_{11}\big{(}{{\bf
x}^{1}}\big{)}&\dots&s_{1r}\big{(}{{\bf x}^{1}}\big{)}\\\ \vdots&&\vdots\\\
s_{r1}\big{(}{{\bf x}^{r}}\big{)}&\dots&s_{rr}\big{(}{\bf
x}^{r}\big{)}\end{matrix}\right),$ (1.8)

where for each $1\leq\alpha\leq r$,

${\bf x}^{\alpha}=\big{(}x^{i_{\alpha}}\big{)},\qquad 1_{\alpha}\leq
i_{\alpha}\leq l_{\alpha}.$

Let $s^{\alpha\beta}$ denote the cofactor of the component $s_{\alpha\beta}$
of $S$. We note that the cofactor $s^{\beta\gamma}$ will not depend on the
group of variables ${\bf x}^{\beta}=\big{(}x^{i_{\beta}}\big{)}$,
$1_{\beta}\leq i_{\beta}\leq l_{\beta}$. Moreover we shall assume that

$\displaystyle\frac{\det S}{s^{\alpha 1}}>0,\qquad\forall\,1\leq\alpha\leq r,$
(1.9)

in order to work with _Riemannian_ Painlevé metrics.

######  Definition 1.1.

Let $S$ be a generalized Stäckel matrix satisfying (1.9). A Painlevé metric is
a Riemannian metric $g$ for which there exist local coordinates ${\bf
x}=\big{(}{\bf x}^{1},\dots,{\bf x}^{r}\big{)}$ such that

$\displaystyle{\rm d}s^{2}=g_{ij}{\rm d}x^{i}{\rm d}x^{j}=\frac{\det
S}{s^{11}}G_{1}+\cdots+\frac{\det S}{s^{r1}}G_{r},$ (1.10)

where each of the quadratic differential forms

$\displaystyle G_{\alpha}=G_{\alpha}({{\bf
x}^{\alpha}})=\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}G_{\alpha}\big{(}{{\bf
x}^{\alpha}}\big{)}_{i_{\alpha}j_{\alpha}}{\rm d}x^{i_{\alpha}}{\rm
d}x^{j_{\alpha}},\qquad 1\leq\alpha\leq r,$ (1.11)

is positive-definite in its arguments and depends only on the group of
variables ${\bf x}^{\alpha}$.

We may thus write the metric (1.10) in block-diagonal form as

${\rm
d}s^{2}=\sum_{\alpha=1}^{r}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}(g_{\alpha})_{i_{\alpha}j_{\alpha}}{\rm
d}x^{i_{\alpha}}{\rm d}x^{j_{\alpha}}=\sum_{\alpha=1}^{r}\frac{\det
S}{s^{\alpha
1}}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}(G_{\alpha})_{i_{\alpha}j_{\alpha}}{\rm
d}x^{i_{\alpha}}{\rm d}x^{j_{\alpha}}.$

It is important to note that even though Painlevé metrics (1.10) are block-
diagonal, and each quadratic differential form (1.11) defines a Riemannian
metric on the submanifolds defined by the level sets ${\bf
x}^{\beta}={\bf{c}}^{\beta}$, $\beta\neq\alpha$, _a Painlevé metric is
generally not a direct sum of Riemannian metrics, nor a warped product, except
for special non-generic cases_. We also note that Painlevé metrics of semi-
Riemannian (and in particular Lorentzian) signature can readily be defined by
modifying the requirement that each of the quadratic differential forms
$G_{\alpha}$ given by (1.11) be positive-definite to one in which
$G_{{\alpha}}$ is assumed to be of signature $(p_{\alpha},q_{\alpha})$ with
$p_{\alpha}+q_{\alpha}=l_{\alpha}$. Finally we also remark that the Painlevé
form (1.10) is obviously invariant under smooth and invertible changes of
coordinates of the form ${\tilde{\bf x}}^{\alpha}={\bf{f}}^{\alpha}({\bf
x}^{\alpha})$, where $1\leq\alpha\leq r$.

Let us call _block orthogonal coordinates_ a system of coordinates $({\bf
x}^{\alpha})$ such that the metric $g$ has the form

$\displaystyle
g=\sum_{\alpha=1}^{r}c_{\alpha}G_{\alpha}=\sum_{\alpha=1}^{r}c_{\alpha}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}(G_{\alpha})_{i_{\alpha}j_{\alpha}}{\rm
d}x^{i_{\alpha}}{\rm d}x^{j_{\alpha}},$

where $c_{\alpha}$ are non-vanishing scalar functions on $M$ and the metrics
$G_{\alpha}$ are given by (1.11). In analogy with the Stäckel case, we have

######  Proposition 1.2.

A metric $g$ is of the Painlevé form (1.10) if and only if there exist block
orthogonal coordinates such that the Hamilton–Jacobi equation

$g^{ij}\partial_{i}W\partial_{j}W=E,$

admits a parametrized family of solutions which is sum-separable into groups
of variables, of the form

$\displaystyle
W=W_{1}\big{(}{\bf{x}}^{1};{{a}}_{1},\dots,{{a}}_{r}\big{)}+\cdots+W_{r}\big{(}{\bf{x}}^{r};{{{a}}_{1},\dots,{a}}_{r}\big{)},$
(1.12)

depending smoothly on $r$ arbitrary real constants
$({{a}}_{1}:=E,a_{2},\dots,{{a}}_{r})$ defined on an open subset
$A\subset\mathbb{R}^{r}$, and satisfying the rank condition

$\displaystyle\det\left(\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\left(\frac{\partial
W_{\alpha}}{\partial
x^{i_{\alpha}}}\right)\left(\frac{\partial^{2}W_{\alpha}}{\partial
a_{\gamma}\partial x^{j_{\alpha}}}\right)\right)\neq 0,$ (1.13)

where

$G^{\alpha}=(G_{\alpha})^{-1}.$

This Proposition will be proved in Section 4 as well as other (intrinsic)
characterizations of Painlevé metrics.

In further analogy with the Stäckel case, we now recall that Painlevé metrics
admit $r$ linearly independent quadratic first integrals of the geodesic flow
which are Poisson commuting. Indeed, the summands $W_{\alpha}$ appearing in
(1.12) satisfy the following set of first-order PDEs [38]

$\displaystyle{\mathcal{F}}_{1}\left({\bf{x}}^{1},\frac{\partial
W_{1}}{\partial{\bf
x}^{1}}\right)=a_{1}s_{11}\big{(}{\bf{x}}^{1}\big{)}+\cdots+a_{r}s_{r1}\big{(}{\bf{x}}^{1}\big{)},$
$\displaystyle\quad\vdots$
$\displaystyle{\mathcal{F}}_{r}\left({\bf{x}}^{r},\frac{\partial
W_{r}}{\partial{\bf
x}^{r}}\right)=a_{1}s_{r1}\big{(}{\bf{x}}^{r}\big{)}+\cdots+a_{r}s_{rr}\big{(}{\bf{x}}^{r}\big{)},$
(1.14)

where

$\displaystyle{\mathcal{F}}_{\alpha}=\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\big{(}{\bf{x}}^{\alpha}\big{)}p_{i_{\alpha}}p_{j_{\alpha}},\qquad
1\leq\alpha\leq r,$ (1.15)

and where the $(a_{\alpha})$ are arbitrary real separation constants. It
follows now directly from the separated equations (1.14) and from the fact
that the generalized Stäckel matrix $S$ is non-singular that one obtains $r$
linearly independent Poisson-commuting quadratic first integrals
$K_{(\alpha)}$ of the geodesic flow by solving for the $r$ separation
constants $(a_{\alpha})$ from the separated equations (1.14). These are
explicitly given by (see [38])

$K_{(\alpha)}=\sum_{\beta=1}^{r}\big{(}S^{-1}\big{)}_{\alpha\beta}\mathcal{F}_{\beta},\qquad
1\leq\beta\leq r,$

or equivalently

$K_{(\alpha)}=K_{(\alpha)}^{ij}p_{i}p_{j},$

where for each $1\leq\alpha\leq r$, $\big{(}K_{(\alpha)}^{ij}\big{)}$ is a
symmetric block-diagonal tensor defined by

$\displaystyle K_{(\alpha)}^{i_{\beta}j_{\beta}}=\frac{s^{\beta\alpha}}{\det
S}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}},\qquad
K_{(\alpha)}^{i_{\beta}j_{\gamma}}=0\qquad\text{for all}\qquad
1\leq\beta\neq\gamma\leq r.$ (1.16)

These quadratic first integrals satisfy

$\displaystyle\big{\\{}K_{(\alpha)},K_{(\beta)}\big{\\}}=0,\qquad
1\leq\alpha,\beta\leq r,\qquad\text{where}\qquad K_{(1)}=H.$ (1.17)

The condition $\\{K_{(\alpha)},H\\}=0$ is equivalent to
$\big{(}K_{(\alpha)}^{ij}\big{)}$ being a symmetric _Killing tensor_ , that is

$\nabla_{i}K_{(\alpha)jk}+\nabla_{j}K_{(\alpha)ki}+\nabla_{k}K_{(\alpha)ij}=0.$

The commutation relations (1.17) are thus equivalent to the vanishing of the
Schouten brackets of the pairs of Killing tensors
$(K_{(\alpha)ij}),(K_{(\beta)ij})$.

There exist a few classical examples of Painlevé metrics in the litterature.
They include for instance the _di Pirro metrics_ [17, 38], for which the
Hamiltonian of the geodesic flow is of the form

$\displaystyle
H=g^{ij}p_{i}p_{j}=\big{(}c_{12}\big{(}x^{1},x^{2}\big{)}+c_{3}\big{(}x^{3}\big{)}\big{)}^{-1}\big{[}a_{1}\big{(}x^{1},x^{2}\big{)}p_{1}^{2}+a_{2}\big{(}x^{1},x^{2}\big{)}p_{2}^{2}+a_{3}\big{(}x^{3}\big{)}p_{3}^{2}\big{]}.$
(1.18)

It may indeed be verified directly that the function

$K=\big{(}c_{12}\big{(}x^{1},x^{2}\big{)}+c_{3}\big{(}x^{3}\big{)}\big{)}^{-1}\big{[}c_{3}\big{(}x^{3}\big{)}(a_{1}\big{(}x^{1},x^{2}\big{)}p_{1}^{2}+a_{2}\big{(}x^{1},x^{2}\big{)}p_{2}^{2})-c_{12}\big{(}x^{1},x^{2}\big{)}a_{3}\big{(}x^{3}\big{)}p_{3}^{2}\big{]}.$

Poisson-commutes with $H$, and thus defines a Killing tensor, which together
with the metric tensor generates a maximal linearly independent set of Killing
tensors for generic choices of the metric functions $c_{12}$, $a_{1}$,
$a_{2}$, $a_{3}$, $c_{3}$ in (1.18). Painlevé metrics also appear in the
context of geodesically equivalent metrics as metrics admitting projective
symmetries, see [43, 44], and also as instances of 4-dimensional Lorentzian
metrics admitting a Killing tensor [23] (see Section 7 for further remarks on
the latter point). Finally, we mention the recent paper by Chanu and Rastelli
[10] that provides a classification of Painlevé metrics with vanishing Riemann
tensor in dimension $3$, i.e., in $\mathbb{E}_{3}$. We will give some examples
of Painlevé metrics in all dimensions satisfying the generalized Robertson
conditions (see below) as well as a catalogue of such metrics in dimensions
$2$, $3$, $4$ in Section 2.

As we stated above, our main goal in this paper is to investigate for the
class of Painlevé metrics the closely related question of product separability
for the Helmholtz equation (1.6), and the relationship between quadratic first
integrals of the geodesic flow and symmetry operators for the Laplace–Beltrami
equation. The Laplace–Beltrami operator for a Painlevé metric $g$ given by
(1.10) can be expressed in terms of the generalized Stäckel matrix $S$ and the
Laplace–Beltrami operators for the $r$ Riemannian metrics $G_{\beta}$,
$1\leq\beta\leq r$, defined by (1.11), corresponding to the blocks of
variables ${\bf x}^{\beta}$, $1\leq\beta\leq r$. We have

$\displaystyle\Delta_{g}u=\sum_{\beta=1}^{r}\Bigg{[}\left(\frac{s^{\beta
1}}{\det S}\right)\Bigg{\\{}\Delta_{G_{\beta}}u$
$\displaystyle\hphantom{\Delta_{g}u=}{}+\sum_{i_{\beta}=1_{\beta}}^{l_{\beta}}\sum_{{j}_{\beta}=1_{\beta}}^{l_{\beta}}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}}\left[\partial_{i_{\beta}}\left(\log\frac{(\det
S)^{\frac{n}{2}-1}s^{\beta
1}}{\big{(}s^{11}\big{)}^{\frac{l_{1}}{2}}\cdots\big{(}s^{r1}\big{)}^{\frac{l_{r}}{2}}}\right)\right]\partial_{j_{\beta}}u\Bigg{\\}}\Bigg{]},$
(1.19)

where $\Delta_{G_{\beta}}$ denotes the Laplace–Beltrami operator for the
Riemannian metric $G_{\beta}$, that is

$\displaystyle\Delta_{G_{\beta}}=\sum_{i_{\beta}=1}^{l_{\beta}}\sum_{j_{\beta}=1}^{l_{\beta}}\frac{1}{\sqrt{|G_{\beta}|}}\partial_{i_{\beta}}\Big{(}\sqrt{|G_{\beta}|}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}}\partial_{j_{\beta}}\Big{)},\qquad|G_{\beta}|=\det((G_{\beta})_{i_{\beta}j_{\beta}}).$

We now state our main results, the proofs of which will be given in Section 6.
We first define the _generalized Robertson conditions_ , in analogy with the
classical Robertson conditions (1.7) for Stäckel metrics.

######  Definition 1.3.

A Painlevé metric $g$ is said to satisfy the generalized Robertson conditions
if and only if the differential conditions

$\displaystyle\partial_{j_{\alpha}}\gamma_{i_{\beta}}=0,\quad\textrm{for
all}\quad 1\leq\alpha\neq\beta\leq r,\quad 1_{\alpha}\leq i_{\alpha}\leq
l_{\alpha},\quad 1_{\beta}\leq i_{\beta}\leq l_{\beta},$ (1.20)

where

$\gamma_{i_{\beta}}=-\partial_{i_{\beta}}\left[\log\frac{(\det
S)^{\frac{n}{2}-1}s^{\beta
1}}{\big{(}s^{11}\big{)}^{\frac{l_{1}}{2}}\cdots\big{(}s^{r1}\big{)}^{\frac{l_{r}}{2}}}\right],$

are satisfied.

The generalized Robertson conditions (1.20) imply that

$\displaystyle\partial_{i_{\alpha}}\gamma^{j_{\beta}}=0,\quad\textrm{for
all}\quad 1\leq\alpha\neq\beta\leq r,$ (1.21)
$\displaystyle\gamma^{j_{\beta}}:=\sum_{i_{\beta}=1_{\beta}}^{l_{\beta}}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}}\gamma_{i_{\beta}}.$
(1.22)

We shall be working with both the forms (1.20) and (1.21) of these conditions.
Note that if the Robertson conditions hold, then the positive Laplace–Beltrami
operator can be written in a synthetic form as

$\displaystyle-\Delta_{g}u=\sum_{\beta=1}^{r}\left(\frac{s^{\beta 1}}{\det
S}\right)\left\\{-\Delta_{G_{\beta}}u+\sum_{{j}_{\beta}=1_{\beta}}^{l_{\beta}}\gamma^{j_{\beta}}\partial_{j_{\beta}}u\right\\}=\sum_{\beta=1}^{r}\left(\frac{s^{\beta
1}}{\det S}\right)B_{\beta}u,$

where the operators
$B_{\beta}=-\Delta_{G_{\beta}}+\sum\limits_{{j}_{\beta}=1_{\beta}}^{l_{\beta}}\gamma^{j_{\beta}}\partial_{j_{\beta}}$
only depend on the group of variables $\textbf{x}^{\beta}$.

As will be seen in Section 5, the generalization of the notion of complete
multiplicative separation for the Helmholtz equation to the case of separation
in terms of groups of variables is given by considering a parametrized family
of product-separable solutions of the form

$\displaystyle u=\prod_{\beta=1}^{r}u_{\beta}\big{(}{\bf
x}^{\beta};a_{1},\dots,a_{r}\big{)},$ (1.23)

satisfying the rank condition

$\displaystyle\det\left(\partial_{a_{\alpha}}\left(\frac{B_{\beta}u_{\beta}}{u_{\beta}}\right)\right)\neq
0,$ (1.24)

where we assume that $u_{\beta}\neq 0$.

Our first result states the separability conditions for the Helmholtz
equation, and gives their interpretation in terms of the vanishing of the off-
block diagonal components of the Ricci tensor:

######  Theorem 1.4.

  1. $1)$

Given a Painlevé metric $g$ of the form (1.10) satisfying the Robertson
conditions (1.20), the Helmholtz equation

$\displaystyle-\Delta_{g}u=\lambda u,$ (1.25)

where $\Delta_{g}$ denotes the Laplace–Beltrami operator (1.19) admits a
solution that is product-separable in the $r$ groups of variables
$\big{(}{{\bf x}^{1}},\dots,{{\bf x}^{r}}\big{)}$,

$\displaystyle u=\prod_{\beta=1}^{r}u_{\beta}\big{(}{\bf
x}^{\beta};a_{1}:=\lambda,a_{2},\dots,a_{r}\big{)},$ (1.26)

and satisfies the rank condition (1.24).

  2. $2)$

The conditions (1.20) may be written equivalently in terms of the Ricci tensor
of the Painlevé metric (1.10) as

$\displaystyle R_{j_{\alpha}k_{\beta}}=0,\quad{\mbox{for all}}\quad
1\leq\alpha\neq\beta\leq r,\quad{\mbox{and}}\quad 1_{\alpha}\leq
j_{\alpha}\leq l_{\alpha},\quad 1_{\beta}\leq k_{\beta}\leq l_{\beta}.$ (1.27)

Our next result shows that the Laplace–Beltrami operator for a Painlevé metric
satisfying the generalized Robertson conditions admits $r$ linearly
independent mutually commuting symmetry operators:

######  Theorem 1.5.

Consider a Painlevé metric (1.10) for which the generalized Robertson
conditions (1.20), which imply the separability of the Helmholtz equation, are
satisfied. Then the operators $\Delta_{K_{\alpha}}$ defined for
$2\leq\alpha\leq r$ by

$\displaystyle\Delta_{K_{(\alpha)}}=\nabla_{i}(K_{(\alpha)}^{ij}\nabla_{j})=\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\nabla_{i_{\gamma}}\big{(}K_{(\alpha)}^{i_{\gamma}j_{\gamma}}\nabla_{j_{\gamma}}\big{)},$
(1.28)

where $K_{(\alpha)}$ is defined by (1.16), commute with the Laplace–Beltrami
operator $\Delta_{g}$ and pairwise commute

$\displaystyle\big{[}\Delta_{K_{(\alpha)}},\Delta_{g}\big{]}=0,\qquad\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=0,\qquad
2\leq\alpha,\beta\leq r,$ (1.29)

and admit the separable solutions (1.26) as formal eigenfunctions with the
separation constants $a_{\alpha}$ arising from the separation of variables as
eigenvalues,

$\Delta_{K_{(\alpha)}}u=a_{\alpha}u,\qquad 2\leq\alpha\leq r.$

Our final result shows that the above framework can be expanded just as in the
Stäckel case by considering conformal deformations of the Painlevé metrics
(1.10) which are compatible with the separation of the Helmholtz equation into
groups of variables. This corresponds to a generalization of the important
notion of $R$-separability [4, 8, 27] to the context of Painlevé metrics.

Let us first recall that upon a conformal rescaling of the metric given by

$\displaystyle g\mapsto c^{4}g,$ (1.30)

where $c$ denotes a smooth positive function, the Laplace–Beltrami operator
$\Delta_{g}$ obeys the transformation law

$\displaystyle\Delta_{c^{4}g}=c^{-(n+2)}\big{(}\Delta_{g}-q_{c,g}\big{)}c^{n-2},$
(1.31)

where

$\displaystyle q_{c,g}=c^{-n+2}\Delta_{g}c^{n-2}.$ (1.32)

Thus, letting

$\displaystyle v=c^{n-2}u$ (1.33)

and using the expression (1.19) of the Laplace–Beltrami operator for a
Painlevé metric $g$, the Helmholtz equation

$\displaystyle-\Delta_{c^{4}g}u=\lambda u,$ (1.34)

takes the form

$\displaystyle\left[\sum_{\beta=1}^{r}\frac{s^{\beta 1}}{\det
S}\left(-\Delta_{G_{\beta}}+\sum_{j_{\beta=1_{\beta}}}^{l_{\beta}}\gamma^{j_{\beta}}\partial_{j_{\beta}}\right)+q_{c,g}-\lambda
c^{4}\right]v=0.$ (1.35)

We have

######  Theorem 1.6.

Let $g$ be a Painlevé metric. Suppose furthermore that $g$ is conformally
rescaled by a factor $c^{4}$ as in (1.30), where $c$ is chosen so as to
satisfy the non-linear PDE

$\displaystyle\Delta_{g}c^{n-2}-\lambda
c^{n+2}-\left(a_{1}+\sum_{\beta=1}^{r}\frac{s^{\beta 1}}{\det
S}(P_{\beta}-\phi_{\beta})\right)c^{n-2}=0,$ (1.36)

where

$\displaystyle
P_{\beta}:=-\frac{1}{2}\partial_{j_{\beta}}\gamma^{j_{\beta}}-\frac{1}{4}\gamma^{j_{\beta}}\partial_{j_{\beta}}\log|G_{\beta}|+\frac{1}{4}(G_{\beta})_{i_{\beta}j_{\beta}}\gamma^{i_{\beta}}\gamma^{j_{\beta}},$

and where $a_{1}$ is a constant and $\phi_{\beta}=\phi_{\beta}\big{(}{{\bf
x}}^{\beta}\big{)}$ are arbitrary smooth functions. Then the Helmholtz
equation (1.34) for the conformally rescaled metric $c^{4}g$ is $R$-separable
in the $r$ groups of variables $\big{(}{{\bf x}^{1}},\dots,{{\bf
x}^{r}}\big{)}$. More precisely, if $u$ is given by

$u=c^{-n+2}Rw,$

with $R$ defined by

$\displaystyle
R=\frac{\big{(}s^{11}\big{)}^{\frac{l_{1}}{4}}\cdots\big{(}s^{r1}\big{)}^{\frac{l_{r}}{4}}}{(\det
S)^{\frac{n-2}{4}}}.$ (1.37)

Then $w$ satisfies

$\displaystyle\sum_{\beta=1}^{r}\left(\frac{s^{\beta 1}}{\det
S}\right)\big{[}{-}\Delta_{G_{\beta}}+\phi_{\beta}\big{]}w=a_{1}w,$ (1.38)

which is separable in the $r$ groups of variables $\big{(}{{\bf
x}^{1}},\dots,{{\bf x}^{r}}\big{)}$ in the sense of (1.23)–(1.24) with the
operators $B_{\beta}$ replaced by the operators
$A_{\beta}=-\Delta_{G_{\beta}}+\phi_{\beta}$.

We remark that the PDE of Yamabe type given by (1.31) satisfied by the
conformal factors $c({\bf x})$ can be viewed as an extension of the
generalized Robertson conditions to the setting of metrics that are
_conformally_ Painlevé. Moreover, the existence of such conformal factors will
be addressed in Section 2 through Proposition 2.4. In particular, it will be
shown there that such metrics enlarge considerably the class of Painlevé
metrics satisfying the generalized Robertson conditions (1.20).

We conclude this section by referring to the interesting recent paper by Chanu
and Rastelli [10] that was published during the elaboration of the present
paper. It turns out that Chanu and Rastelli define the Painlevé form of
metrics like our Definition 1.1 in connection with the notion of separability
of the Hamilton–Jacobi equations in groups of variables. They provide several
intrinsic characterizations of Painlevé metrics extending the ones stated in
our Section 5. We refer for instance to the beautiful invariant
characterization of Painlevé metrics given in their Proposition 5.8 that allow
them to classify all Painlevé metrics in $\mathbb{E}_{3}$.

##  2 Examples of Painlevé metrics satisfying
the generalized Robertson conditions

In this section, we provide several examples of Painlevé metrics satisfying
the generalized Robertson conditions (1.20) in all dimensions. Then we try to
give a catalogue – as complete as possible – of such Painlevé metrics in
dimensions $2$, $3$ and $4$. All our examples are local in the sense that they
are defined in a single coordinate chart. Obtaining global examples of
Riemannian or semi-Riemannian manifolds admitting an atlas of coordinate
charts in which the metric is in Painlevé form appears to be a challenging
task, well worthy of further investigation. This point will be discussed as
one of the perspectives listed in Section 7. From the notations used in
definition (1.1), recall that a Painlevé metric is given in local coordinates
$\big{(}x^{1},\dots,x^{n}\big{)}=\big{(}{\bf x}^{1},\dots,{\bf x}^{r}\big{)}$
where ${\bf x}^{\alpha}$ denotes group of variables indexed by
$1\leq\alpha\leq r$ by

$\displaystyle g=g_{ij}{\rm d}x^{i}{\rm d}x^{j}=\frac{\det
S}{s^{11}}G_{1}+\cdots+\frac{\det S}{s^{r1}}G_{r},$

for quadratic differential forms

$\displaystyle G_{\alpha}=G_{\alpha}\big{(}{{\bf
x}^{\alpha}}\big{)}=\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}G_{\alpha}\big{(}{{\bf
x}^{\alpha}}\big{)}_{i_{\alpha}j_{\alpha}}{\rm d}x^{i_{\alpha}}{\rm
d}x^{j_{\alpha}},\qquad 1\leq\alpha\leq r,$ (2.1)

and a generalized Stäckel matrix $S$ of the form

$S=\left(\begin{matrix}s_{11}\big{(}{{\bf
x}^{1}}\big{)}&\dots&s_{1r}\big{(}{{\bf x}^{1}}\big{)}\\\ \vdots&&\vdots\\\
s_{r1}\big{(}{{\bf x}^{r}}\big{)}&\dots&s_{rr}\big{(}{\bf
x}^{r}\big{)}\end{matrix}\right).$

From (1.20), recall also that the Robertson conditions read

$\partial_{j_{\alpha}}\gamma_{i_{\beta}}=0,\qquad 1\leq\alpha\neq\beta\leq
r,\quad 1_{\alpha}\leq i_{\alpha}\leq l_{\alpha},\quad 1_{\beta}\leq
i_{\beta}\leq l_{\beta},$

where

$\gamma_{i_{\beta}}=-\partial_{i_{\beta}}\left[\log\frac{(\det
S)^{\frac{n}{2}-1}s^{\beta
1}}{\big{(}s^{11}\big{)}^{\frac{l_{1}}{2}}\cdots\big{(}s^{r1}\big{)}^{\frac{l_{r}}{2}}}\right].$

Since $s^{\beta 1}$ does not depend on the group of variables
$\bf{x^{\beta}}$, these conditions can be equivalently formulated as the
algebraic-differential condition

$\displaystyle\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}=\prod_{\alpha=1}^{r}f_{\alpha}\big{(}\bf{x^{\alpha}}\big{)},$
(2.2)

where $f_{\alpha}=f_{\alpha}\big{(}\bf{x^{\alpha}}\big{)}$ are arbitrary
functions of the indicated group of variables. We will use this last
expression of the Robertson conditions to find different examples of Painlevé
metrics in all dimensions that satisfy them. Our main examples are:

######  Example 2.1.

If $r=2$ and $n=2$, then any Stäckel matrix

$S=\left(\begin{matrix}s_{11}\big{(}x^{1}\big{)}&s_{12}\big{(}x^{1}\big{)}\\\
s_{21}\big{(}x^{2}\big{)}&s_{22}\big{(}x^{2}\big{)}\end{matrix}\right),$

satisfies automatically the usual Robertson conditions (2.2). The
corresponding Stäckel metrics in 2D can be given the following _normal form_

$\displaystyle
g=\big{(}f_{1}\big{(}x^{1}\big{)}+f_{2}\big{(}x^{2}\big{)}\big{)}\big{(}\big{(}{\rm
d}x^{1}\big{)}^{2}+\big{(}{\rm d}x^{2}\big{)}^{2}\big{)},$

where $f_{\alpha}$, $\alpha=1,2$ are arbitrary functions of ${\bf x}^{\alpha}$
such that $f_{1}+f_{2}>0$. Thus we recover the classical Liouville metrics.

If $r=2$ and $n\geq 3$, then any generalized Stäckel matrix

$S=\left(\begin{matrix}s_{11}\big{(}{\bf x}^{1}\big{)}&s_{12}\big{(}{\bf
x}^{1}\big{)}\\\ 0&s_{22}\big{(}{\bf x}^{2}\big{)}\end{matrix}\right),$

satisfy the generalized Robertson conditions (2.2). The corresponding Painlevé
metrics can be given the following _normal form_

$\displaystyle g=G_{1}+f_{1}\big{(}\textbf{x}^{1}\big{)}G_{2},$

where $G_{1}$, $G_{2}$ are Riemannian metrics as in (2.1) and
$f_{1}=f_{1}\big{(}\textbf{x}^{1}\big{)}$ is any positive function. Note that
the metrics are classical _warped products_.

######  Example 2.2.

Consider a generalized Stäckel matrix of the form

$S=\left(\begin{matrix}s_{11}\big{(}{{\bf
x}^{1}}\big{)}&\dots&s_{1r}\big{(}{{\bf x}^{1}}\big{)}\\\
a_{21}&\dots&a_{2r}\\\ \vdots&&\vdots\\\
a_{r1}&\dots&a_{rr}\end{matrix}\right),$

where the entries $a_{\alpha\beta}$, $2\leq\alpha\leq r$, $1\leq\beta\leq r$,
are real constants chosen such that (1.9) is satisfied. Then it is immediate
that

$\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}=f_{1}\big{(}{\bf
x}^{1}\big{)},$

for some function $f_{1}$ depending only on ${\bf x}^{1}$. Hence the Robertson
conditions (2.2) are trivially satisfied. The corresponding Painlevé metrics
are of the general form of _multiply warped products_

$\displaystyle g=\sum_{\alpha=1}^{r}f_{\alpha}\big{(}{\bf
x}^{1}\big{)}G_{\alpha},$ (2.3)

where $f_{\alpha}$ are arbitrary positive functions of ${\bf x}^{1}$ and
$G_{\alpha}$ are given by (2.1). We note that the inverse anisotropic Calderón
problem on a class of singular metrics of the form (2.3) is studied in [14].

######  Example 2.3.

Our final class of examples is the most interesting one and comes from the
theory of geodesically (or projectively) equivalent metrics (see for instance
[6, 34, 43, 44, 45]) and its link to particular Stäckel systems called Benenti
systems [1, 5]. Note that it only applies to Stäckel metrics satisfying the
usual Robertson conditions, i.e., we assume that $r=n$ in the following. Note
also that in the context of geodesically equivalent metrics, the commutation
relations of Theorem 1.5 were already established by direct computation in
[35], and can also be seen to follow from the commutation of the corresponding
Killing tensors with the Ricci tensor [30]. Consider a Stäckel matrix $S$ of
Vandermonde type (see [1, Theorem 8.5])

$S=\big{(}(-1)^{n+\alpha-\beta+1}f_{\alpha}^{n-\beta}\big{)}_{1\leq\alpha,\beta\leq
n},$

where the functions $f_{\alpha}=f_{\alpha}(x^{\alpha})$ only depend on the
variable $x^{\alpha}$ and satisfy

$f_{1}\big{(}x^{1}\big{)}<f_{2}\big{(}x^{2}\big{)}<\dots<f_{r}\big{(}x^{r}\big{)},\qquad\forall\,x^{\alpha},\quad
1\leq\alpha\leq n.$

An easy calculation shows that

$\det(S)=\prod_{1\leq\alpha<\beta\leq n}|f_{\alpha}-f_{\beta}|,\qquad
s^{\gamma 1}=\prod_{\begin{subarray}{c}1\leq\alpha<\beta\leq n\\\
\alpha,\beta\neq\gamma\end{subarray}}|f_{\alpha}-f_{\beta}|,\qquad\forall\,1\leq\gamma\leq
n,$

from which we deduce that the Robertson conditions (2.2) are satisfied. The
corresponding Stäckel metrics are given by

$\displaystyle g=\left(\prod_{\alpha\neq
1}|f_{\alpha}-f_{1}|\right)\big{(}{\rm
d}x^{1}\big{)}^{2}+\left(\prod_{\alpha\neq
2}|f_{\alpha}-f_{2}|\right)\big{(}{\rm
d}x^{2}\big{)}^{2}+\dots+\left(\prod_{\alpha\neq
n}|f_{\alpha}-f_{n}|\right)\big{(}{\rm d}x^{n}\big{)}^{2}.$

Let us now use the above classes of examples to give as exhaustively as
possible a list of Painlevé metrics satisfying the generalized Robertson
conditions in dimensions $n=2,3,4$. Note that the list of Painlevé metrics
given below is only exhaustive as far as generic cases are concerned, and thus
does not cover all the examples of Painlevé metrics satisfying the Robertson
conditions, such as metrics of constant sectional curvature. Recall also that
we always assume that $2\leq r\leq n$.

2D Painlevé metrics. Let $n=2$ and $r=2$. Then according to Example 2.1, the
only Painlevé metrics are Stäckel metrics given by

$\displaystyle
g=\big{(}f_{1}\big{(}x^{1}\big{)}+f_{2}\big{(}x^{2}\big{)}\big{)}\big{(}\big{(}{\rm
d}x^{1}\big{)}^{2}+\big{(}{\rm d}x^{2}\big{)}^{2}\big{)},$

for some functions $f_{1}$ and $f_{2}$ such that $f_{1}+f_{2}>0$. Hence
Painlevé metrics satisfying the Robertson conditions in 2D are Liouville
metrics.

3D Painlevé metrics. Let $n=3$.

  * •

If $r=2$ and say $l_{1}=1$, $l_{2}=2$, then according to Example 2.1, Painlevé
metrics satisfying the generalized Robertson conditions are classical warped
products; more precisely

$\displaystyle g=\big{(}{\rm
d}x^{1}\big{)}^{2}+f_{1}\big{(}x^{1}\big{)}G_{2},\qquad\textrm{or}\qquad
g=f_{2}\big{(}\textbf{x}^{2}\big{)}\big{(}{\rm d}x^{1}\big{)}^{2}+G_{2},$

for some positive functions $f_{1}$ and $f_{2}$ depending only on the
indicated groups of variables and any Riemannian metric

$G_{2}=(G_{2})_{ij}\big{(}x^{2},x^{3}\big{)}{\rm d}x^{i}{\rm d}x^{j},\qquad
i,j=2,3.$

  * •

If $r=3$, then 3D Painlevé metrics are in fact Stäckel metrics. According to
Examples 2.2 and 2.3, we have the following possible expressions for Stäckel
metrics $g$ satisfying the Robertson conditions (see also [21]):

$\displaystyle g=f_{1}\big{(}{\rm d}x^{1}\big{)}^{2}+h_{1}\big{(}{\rm
d}x^{2}\big{)}^{2}+k_{1}\big{(}{\rm d}x^{3}\big{)}^{2},$

or

$\displaystyle g=(f_{3}-f_{1})(f_{2}-f_{1})\big{(}{\rm
d}x^{1}\big{)}^{2}+(f_{3}-f_{2})(f_{2}-f_{1})\big{(}{\rm
d}x^{2}\big{)}^{2}+(f_{3}-f_{1})(f_{3}-f_{2})\big{(}{\rm d}x^{3}\big{)}^{2},$

where $f_{1}$, $h_{1}$, $k_{1}$ are functions of the variable $x^{1}$ only and
$f_{2}$, $f_{3}$ are functions of the variables $x^{2}$ and $x^{3}$
respectively such that $f_{1}<f_{2}<f_{3}$. We add a last example to this list
found by inspection of the Robertson conditions (2.2). Consider the Stäckel
matrix

$S=\left(\begin{matrix}1&s_{12}&as_{13}\\\ 0&s_{22}&s_{23}\\\
0&s_{32}&s_{33}\end{matrix}\right),$

where $a$ is a real constant and the $s_{ij}=s_{ij}\big{(}x^{i}\big{)}$ are
arbitrary functions of the indicated variables for which $\det S\neq 0$. Then
we can check directly that the Robertson conditions are satisfied and we
obtain the following expression for the corresponding Stäckel metrics

$\displaystyle g=\big{(}{\rm
d}x^{1}\big{)}^{2}+\left(\frac{1}{s_{12}}\right)\left[(s_{22}s_{33}-s_{23}s_{32})\left(\frac{\big{(}{\rm
d}x^{2}\big{)}^{2}}{s_{32}-as_{33}}+\frac{\big{(}{\rm
d}x^{3}\big{)}^{2}}{s_{23}-as_{22}}\right)\right].$ (2.4)

Note in particular that such metrics are warped products and thus admit a
conformal Killing vector field.

4D Painlevé metrics. Let $n=4$.

  * •

If $r=2$ and $l_{1}+l_{2}=4$, then according to Example 2.1, Painlevé metrics
that satisfy the generalized Robertson conditions are warped products of the
type

$\displaystyle
g=G_{1}+f_{1}\big{(}\textbf{x}^{1}\big{)}G_{2},\qquad\textrm{or}\qquad
g=f_{2}\big{(}\textbf{x}^{2}\big{)}G_{1}+G_{2},$

for some positive functions $f_{1}$ and $f_{2}$ depending only on the
indicated variables and any Riemannian metrics $G_{1}$, $G_{2}$ of the type
(2.1).

  * •

If $r=3$ and $l_{1}=2$, $l_{2}=l_{3}=1$ (the other cases are treated
similarly), then according to Example 2.2, we obtain the following Painlevé
metrics

$\displaystyle g=hG_{1}+k\big{(}{\rm d}x^{3}\big{)}^{2}+l\big{(}{\rm
d}x^{4}\big{)}^{2},$

where $h$, $k$, $l$ are positive functions of the variables $x^{1}$, $x^{2}$
only and

$\displaystyle G_{1}=(G_{1})_{ij}\big{(}x^{1},x^{2}\big{)}{\rm d}x^{i}{\rm
d}x^{j},\qquad i,j=1,2$

is any Riemannian metric. Following the same procedure as in example (2.4), we
also obtain the following class of Painlevé metrics

$\displaystyle
g=G_{1}+\left(\frac{1}{s_{12}}\right)\left[(s_{22}s_{33}-s_{23}s_{32})\left(\frac{\big{(}{\rm
d}x^{3}\big{)}^{2}}{s_{32}-as_{33}}+\frac{\big{(}{\rm
d}x^{4}\big{)}^{2}}{s_{23}-as_{22}}\right)\right],$

where $s_{12}=s_{12}\big{(}x^{1},x^{2}\big{)}$,
$s_{22}=s_{22}\big{(}x^{3}\big{)}$, $s_{23}=s_{23}\big{(}x^{3}\big{)}$,
$s_{32}=s_{32}\big{(}x^{4}\big{)}$, $s_{33}=s_{33}\big{(}x^{4}\big{)}$ and
$G_{1}=(G_{1})_{ij}\big{(}x^{1},x^{2}\big{)}{\rm d}x^{i}{\rm d}x^{j}$,
$i,j=1,2$. Note in particular that such metrics are warped products that admit
a conformal Killing vector field.

  * •

If $r=4$, the Painlevé metrics are Stäckel metrics. According to Examples 2.2
and 2.3, possible expressions for Painlevé metrics satisfying the Robertson
conditions are

$\displaystyle g=f_{1}\big{(}{\rm d}x^{1}\big{)}^{2}+h_{1}\big{(}{\rm
d}x^{2}\big{)}^{2}+k_{1}\big{(}{\rm d}x^{3}\big{)}^{2}+l_{1}\big{(}{\rm
d}x^{4}\big{)}^{2},$

or

$\displaystyle g=(f_{4}-f_{1})(f_{3}-f_{1})(f_{2}-f_{1})\big{(}{\rm
d}x^{1}\big{)}^{2}+(f_{4}-f_{2})(f_{3}-f_{2})(f_{2}-f_{1})\big{(}{\rm
d}x^{2}\big{)}^{2}$
$\displaystyle\hphantom{g=}{}+(f_{4}-f_{3})(f_{3}-f_{1})(f_{3}-f_{2})\big{(}{\rm
d}x^{3}\big{)}^{2}+(f_{4}-f_{1})(f_{4}-f_{2})(f_{4}-f_{3})\big{(}{\rm
d}x^{4}\big{)}^{2},$

where $f_{1}$, $h_{1}$, $k_{1}$, $l_{1}$ are functions of the variable $x^{1}$
only and $f_{2}$, $f_{3}$, $f_{4}$ are functions of the variables $x^{2}$,
$x^{3}$ and $x^{4}$ respectively such that $f_{1}<f_{2}<f_{3}<f_{4}$. We add a
last example to this list found by inspection of the Robertson conditions
(2.2). Consider the Stäckel matrix

$S=\left(\begin{matrix}1&s_{12}&as_{12}&s_{12}\\\ 0&1&s_{23}&s_{23}\\\
0&0&s_{33}&s_{34}\\\ 0&0&s_{43}&s_{44}\end{matrix}\right),$

where $s_{ij}=s_{ij}\big{(}x^{i}\big{)}$ arbitrary functions of the indicated
variables. Then we can check directly that the generalized Robertson
conditions are satisfied and we obtain the following expression for the
corresponding Stäckel metrics

$\displaystyle g=\big{(}{\rm
d}x^{1}\big{)}^{2}+\left(\frac{1}{s_{12}}\right)\Bigg{[}\big{(}{\rm
d}x^{2}\big{)}^{2}$
$\displaystyle\hphantom{g=}{}+\left(\frac{1}{s_{23}-1}\right)\left((s_{33}s_{44}-s_{34}s_{43})\left(\frac{\big{(}{\rm
d}x^{3}\big{)}^{2}}{s_{44}-s_{43}}+\frac{\big{(}{\rm
d}x^{4}\big{)}^{2}}{s_{33}-s_{34}}\right)\right)\Bigg{]}.$

Note that such metrics are warped products and that the metrics between square
brackets are also warped products.

We end this section by giving some existence results for the conformal factor
$c(\bf x)$ appearing in Theorem 1.6, in the case in which $M$ is a smooth
compact manifold of dimension $n\geq 3$, with smooth boundary $\partial M$. We
recall from Theorem 1.6 that the conformal factor $c(\bf x)$ must satisfy a
non-linear PDE of Yamabe type, given by

$\displaystyle\Delta_{g}c^{n-2}+f({\bf x})c^{n-2}-\lambda c^{n+2}=0,$

where

$\displaystyle f({\bf x})=\left(\sum_{\beta=1}^{r}\frac{s^{\beta 1}}{\det
S}(\phi_{\beta}-P_{\beta})\right)-a_{1},$

and where $\phi_{\beta}=\phi_{\beta}\big{(}{{\bf x}}^{\beta}\big{)}$ are
arbitrary smooth functions. Setting $w=c^{n-2}$, we are thus interested in
solutions $w=c^{n-2}$ of the non-linear elliptic PDE:

$\displaystyle\Delta_{g}w+f({\bf x})w-\lambda w^{\frac{n+2}{n-2}}=0,\qquad$
$\displaystyle\textrm{on}\ M,$ $\displaystyle w=\eta,$
$\displaystyle\textrm{on}\ \partial M,$ (2.5)

where $\eta$ is any suitable smooth positive function on $\partial M$. We can
solve (2.5) by using the well-known technique of lower and upper solutions
which we briefly recall here. Setting

$f({\bf x},w)=f({\bf x})w-\lambda w^{\frac{n+2}{n-2}},$

we recall that an upper solution ${\overline{w}}$ is a function in
$C^{2}(M)\cap C^{0}(\overline{M})$ satisfying

$\displaystyle\Delta_{g}{\overline{w}}+f({\bf x},{\overline{w}})\leq
0\quad\textrm{on}\ M,\qquad\textrm{and}\qquad{\overline{w}}_{|\partial
M}\geq\eta.$

Similarly, a lower solution ${\underline{w}}$ is a function in $C^{2}(M)\cap
C^{0}(\overline{M})$ satisfying

$\displaystyle\Delta_{g}{\underline{w}}+f({\bf x},{\underline{w}})\geq
0\quad\textrm{on}\ M,\qquad\textrm{and}\qquad{\underline{w}}_{|\partial
M}\leq\eta.$

It is well-known that if we can find a lower solution ${\underline{w}}$ and an
upper solution ${\overline{w}}$ satisfying ${\underline{w}}\leq{\overline{w}}$
on $M$, then there exists a solution $w\in C^{\infty}(\overline{M})$ of (2.5)
such that ${\underline{w}}\leq w\leq{\overline{w}}$ on $M$.

Now, we can prove the following result:

######  Proposition 2.4.

  1. $1.$

If $\lambda>0$ and $f({\bf x})>0$ on $M$, then for each positive function
$\eta$ on $\partial M$, there exists a smooth positive solution $w$ of (2.5).

  2. $2.$

If $\lambda\leq 0$ and $f({\bf x})<\lambda$ on $M$, then for each for each
positive function $\eta$ on $\partial M$ such that $\eta\leq 1$, there exists
a smooth positive solution $w$ of (2.5).

######  Remark 2.5.

Since $\frac{s^{\beta 1}}{\det S}>0$ by the hypothesis (1.9), we see that the
assumption $f({\bf x})>0$ on $M$ (resp. $f({\bf x})<\lambda$ on $M$) is
satisfied if the $\phi_{\beta}$’s are chosen sufficiently large (resp.
$-\phi_{\beta}$ are sufficiently large).

###### Proof.

1\. We use the technique of lower and upper solutions. We define
${\underline{w}}=\epsilon$ where $\epsilon>0$ is small enough. Thus,
$\underline{w}\leq\eta$ on $\partial M$ and we have

$\Delta_{g}\underline{w}+f({\bf x},\underline{w})=\epsilon\big{(}f({\bf
x})-\lambda\epsilon^{{\frac{n+2}{n-2}}-1}\big{)}>0,$

so $\underline{w}$ is a lower solution. In the same way, we define
$\overline{w}=C$ where $C$ is sufficiently large. Thus $\overline{w}\geq\eta$
and we have

$\Delta_{g}\overline{w}+f({\bf x},\overline{w})=C\big{(}f({\bf x})-\lambda
C^{{\frac{n+2}{n-2}}-1}\big{)}<0.$

It follows that $\overline{w}$ is an upper solution and clearly
$\underline{w}\leq\overline{w}$. Thus, there exist a smooth positive solution
$w$ of (2.5) satisfying $\epsilon\leq w\leq C$.

2\. In the case $\lambda\leq 0$, $f({\bf x})<\lambda$ on $M$ and $\eta\leq 1$
on $\partial M$, we define ${\underline{w}}$ as the unique solution of the
Dirichlet problem

$\displaystyle\Delta_{g}\underline{w}+f({\bf x})\underline{w}=0,\qquad$
$\displaystyle\textrm{on}\ M,$ $\displaystyle\underline{w}=\eta,\qquad$
$\displaystyle\textrm{on}\ \partial M.$

The strong maximum principle implies that $0<\underline{w}\leq\max\eta$ on
$M$. Moreover, $\triangle_{g}\underline{w}+f({\bf
x},\underline{w})=-\lambda(\underline{w})^{\frac{n+2}{n-2}}\geq 0$. Hence
$\underline{w}$ is a lower solution of (2.5).

Now, we define ${\overline{w}}$ as the unique solution of the Dirichlet
problem

$\displaystyle\Delta_{g}\overline{w}+f({\bf x})\overline{w}=f({\bf
x})(\max\eta)^{\frac{n+2}{n-2}},\qquad$ $\displaystyle\textrm{on}\ M,$
$\displaystyle\overline{w}=\eta,$ $\displaystyle\textrm{on}\ \partial M.$

According to the maximum principle, we also have $\overline{w}\geq 0$ on $M$.
Setting $v=\overline{w}-\max\eta$, we see that

$\Delta_{g}v+f({\bf x})v=f({\bf
x})\big{(}\max\eta^{\frac{n+2}{n-2}}-\max\eta\big{)}\geq 0,$

since $\eta\leq 1$. Hence, the maximum principle implies that $v\leq 0$ on
$M$, or equivalently $\overline{w}\leq\max\eta$.

We deduce that

$\Delta_{g}\overline{w}+f({\bf x},\overline{w})=f({\bf
x})\big{(}\max\eta^{\frac{n+2}{n-2}}-\overline{w}^{\frac{n+2}{n-2}}\big{)}+(f({\bf
x})-\lambda)\overline{w}^{\frac{n+2}{n-2}}\leq 0,$

since $f({\bf x})<\lambda$. Thus, $\overline{w}$ is an upper solution of
(2.5). Finally, $\overline{w}-\underline{w}$ satisfies

$\displaystyle\Delta_{g}(\overline{w}-\underline{w})+f({\bf
x})(\overline{w}-\underline{w})=f({\bf
x})(\max\eta)^{\frac{n+2}{n-2}}<0,\qquad$ $\displaystyle\textrm{on}\ M,$
$\displaystyle\overline{w}-\underline{w}=0,$ $\displaystyle\textrm{on}\
\partial M.$

Then, the maximum principle implies again $\overline{w}\geq\underline{w}$.
Then according to the lower and upper solutions technique, there exists a
smooth positive solution $w$ of (2.5). ∎

We conclude by remarking that there exist important classes of $n$-dimensional
metrics of physical interest for which the geodesic flow admits $[n/2]-1$
Poisson-commuting quadratic first integrals arising from the presence of a
principal Killing–Yano tensor with torsion. We refer to [19, 24] for results
on their local classification and normal forms.

##  3 Generalized Killing–Eisenhart and Levi-Civita conditions

The proofs of the main results of our paper, that is Theorems 1.4, 1.5 and
1.6, make use of generalizations to Painlevé metrics of the classical
Killing–Eisenhart equations and Levi-Civita separability conditions which hold
for Stäckel metrics (see for example [2, 3, 18, 25, 26, 38]). We present these
generalizations in the form of the following two lemmas, beginning with the
Killing–Eisenhart equations. Thus in analogy with the Stäckel case, we
introduce the quantities

$\displaystyle\rho_{\beta\gamma}:=\frac{s^{\gamma\beta}}{s^{\gamma 1}}.$ (3.1)

Note that by the assumption (1.9), we have $s^{\gamma 1}\neq 0$. The following
lemma gives the generalization to the case of Painlevé metrics of the
Killing–Eisenhart equations given in [2, 3, 18, 25, 26] for Stäckel metrics:

######  Lemma 3.1.

We have, for all $1\leq\beta,\delta,\gamma\leq r$, the identities

$\displaystyle\partial_{j_{\gamma}}(\rho_{\beta\delta})=(\rho_{\beta\gamma}-\rho_{\beta\delta})\left(\partial_{j_{\gamma}}{\log\frac{s^{\delta
1}}{\det S}}\right).$ (3.2)

We will refer to (3.2) as the _generalized Killing–Eisenhart equations._

###### Proof.

The Poisson bracket relations (1.17) imply

$\displaystyle\sum_{p_{\gamma}=1}^{l_{\gamma}}\big{(}\big{(}\partial_{p_{\gamma}}K_{(1)}^{i_{\delta}j_{\delta}}\big{)}K_{(\beta)}^{p_{\gamma}k_{\gamma}}-\big{(}\partial_{p_{\gamma}}K_{(\beta)}^{i_{\delta}j_{\delta}}\big{)}K_{(1)}^{p_{\gamma}k_{\gamma}}\big{)}=0,$
(3.3)

where $1\leq i_{\delta},j_{\delta}\leq l_{\delta}$, $1\leq k_{\gamma}\leq
l_{\gamma}$ and $1\leq\delta,\gamma\leq r$. Using the expressions (1.16) of
the Killing tensors $\big{(}K_{(\beta)}^{i_{\delta}j_{\delta}}\big{)}$, the
fact that
$\partial_{i_{\gamma}}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}}=0$ for
$\beta\neq\gamma$, and the fact that each of the $l_{\gamma}\times l_{\gamma}$
matrices $\big{(}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{)}$, is
invertible, we obtain that the relations (3.3) are equivalent to

$\displaystyle\left[\partial_{p_{\gamma}}\left(\frac{s^{\delta 1}}{\det
S}\right)\right]\frac{s^{\gamma\beta}}{\det
S}-\left[\partial_{p_{\gamma}}\left(\frac{s^{\delta\beta}}{\det
S}\right)\right]\frac{s^{\gamma 1}}{\det S}=0,$ (3.4)

where $1\leq\delta,\gamma\leq r$, $1_{\gamma}\leq p_{\gamma}\leq l_{\gamma}$.
In particular, the relations (3.3) are independent of the block metrics
(1.11). Setting $\delta=\gamma$ in (3.4), we obtain

$\partial_{p_{\gamma}}\left(\frac{s^{\gamma\beta}}{s^{\gamma 1}}\right)=0,$

for $1_{\gamma}\leq p_{\gamma}\leq l_{\gamma}$, so that using the definition
(3.1) of the quantities $\rho_{\beta\gamma}$, the relations (3.4) take the
form

$\partial_{p_{\gamma}}\left(\rho_{\beta\delta}\frac{s^{\delta 1}}{\det
S}\right)\frac{s^{\gamma 1}}{\det
S}=\left(\partial_{p_{\gamma}}\left(\frac{s^{\delta 1}}{\det
S}\right)\right)\rho_{\beta\gamma}\frac{s^{\gamma 1}}{\det S},$

which in turn reduces to (3.2), thus proving our claim. ∎

In order to state the generalization to Painlevé metrics of the classical
Levi-Civita conditions which hold for Stäckel metrics, we make the hypothesis

$\displaystyle\rho_{\beta\delta}\neq\rho_{\beta\epsilon},\qquad\forall\,1\leq\beta\leq
r,\quad\forall\,1\leq\delta\neq\epsilon\leq r.$ (3.5)

The generalization of the Levi-Civita conditions to the Painlevé case is now
given by the following:

######  Lemma 3.2.

A generalized Stäckel matrix (1.8) corresponding to a Painlevé metric (1.10)
for which the genericity hypothesis (3.5) holds true satisfies, for all
$1\leq\beta,\gamma\leq r$, the identities

$\displaystyle\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma 1}}{\det
S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{s^{\alpha 1}}{\det
S}\right)\right)+\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\beta 1}}{\det
S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma 1}}{\det
S}\right)\right)$ $\displaystyle\qquad{}-\frac{\det S}{s^{\gamma
1}}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{s^{\gamma 1}}{\det
S}\right)=0.$ (3.6)

In particular, we have the identities

$\displaystyle\frac{\partial}{\partial x^{j_{\alpha}}}\frac{\partial}{\partial
x^{k_{\beta}}}\left(\frac{\det S}{s^{\alpha 1}s^{\beta 1}}\right)=0,$ (3.7)

for all $1\leq\alpha,\beta\leq r$.

We will likewise refer to the identities (3.6) as the _generalized Levi-Civita
conditions_.

######  Remark 3.3.

  1. 1)

In the case of Stäckel metrics, that is when $r=n$, the conditions (3.6)
reduce to the classical Levi-Civita conditions, given by

$\displaystyle\left(\partial_{j}\log\left(\frac{s^{l1}}{\det
S}\right)\right)\left(\partial_{k}\log\left(\frac{s^{j1}}{\det
S}\right)\right)+\left(\partial_{j}\log\left(\frac{s^{k1}}{\det
S}\right)\right)\left(\partial_{k}\log\left(\frac{s^{l1}}{\det
S}\right)\right)$ $\displaystyle\qquad{}-\frac{\det
S}{s^{l1}}\partial_{j}\partial_{k}\left(\frac{s^{l1}}{\det S}\right)=0.$

  2. 2)

We shall show at the end of the next Section 4 that the generalized Levi-
Civita conditions (3.6) hold in fact for all Painlevé metrics (1.10) without
assuming our genericity hypothesis (3.5). Nevertheless, it is easier to obtain
them from the Killing–Eisenhart equations under the assumption (3.5) as we do
below.

###### Proof.

The general idea behind the proof is similar to the one that is used in the
classical Stäckel case, and is based on expressing the integrability
conditions for the generalized Killing–Eisenhart (3.2), with additional twist
resulting from the fact that the separation is in groups of variables only. We
let $1\leq\alpha\neq\beta\leq r$ denote fixed indices and introduce the
simplified notation

$\rho_{\delta}:=\rho_{\beta\delta}=\frac{s^{\delta\beta}}{s^{\delta 1}},$

so as to make the expressions a bit more compact. The generalized
Killing–Eisenhart equations (3.2) thus take the form

$\displaystyle\partial_{p_{\gamma}}\rho_{\delta}=(\rho_{\gamma}-\rho_{\delta})\left(\partial_{p_{\gamma}}\left(\log\frac{s^{\delta
1}}{\det S}\right)\right),$

where $1\leq\delta,\gamma\leq r$. The integrability conditions

$\displaystyle\partial_{k_{\epsilon}}\big{(}\partial_{p_{\gamma}}\rho_{\delta}\big{)}=\partial_{p_{\gamma}}\big{(}\partial_{k_{\epsilon}}\rho_{\delta}\big{)}$
(3.8)

are now easily obtained. We have

$\displaystyle\partial_{k_{\epsilon}}(\partial_{p_{\gamma}}\rho_{\delta})=\left[(\rho_{\epsilon}-\rho_{\gamma})\partial_{k_{\epsilon}}\log\left(\frac{s^{\gamma
1}}{\det
S}\right)-(\rho_{\epsilon}-\rho_{\delta})\partial_{k_{\epsilon}}\log\left(\frac{s^{\delta
1}}{\det S}\right)\right]\partial_{p_{\gamma}}\log\frac{s^{\delta 1}}{\det S}$
$\displaystyle\hphantom{\partial_{k_{\epsilon}}(\partial_{p_{\gamma}}\rho_{\delta})=}{}+(\rho_{\gamma}-\rho_{\delta})\partial_{k_{\epsilon}}\partial_{p_{\gamma}}\log\left(\frac{s^{\delta
1}}{\det S}\right),$

so that (3.8) becomes

$\displaystyle(\rho_{\gamma}-\rho_{\epsilon})\left[\left(\partial_{k_{\epsilon}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)\left(\partial_{p_{\gamma}}\log\left(\frac{s^{\delta
1}}{\det S}\right)\right)\right.$
$\displaystyle\qquad{}+\left(\partial_{k_{\epsilon}}\log\left(\frac{s^{\delta
1}}{\det
S}\right)\right)\left(\partial_{p_{\gamma}}\log\left(\frac{s^{\epsilon
1}}{\det S}\right)\right)$
$\displaystyle\left.\qquad{}-\left(\partial_{k_{\epsilon}}\log\left(\frac{s^{\delta
1}}{\det S}\right)\right)\left(\partial_{p_{\gamma}}\log\left(\frac{s^{\delta
1}}{\det
S}\right)\right)-\partial_{k_{\epsilon}}\partial_{p_{\gamma}}\log\left(\frac{s^{\delta
1}}{\det S}\right)\right]=0,$ (3.9)

where $1\leq\epsilon\neq\gamma\leq r$ and $1\leq\delta\leq r$. Using the rank
hypothesis (3.5), expanding the logarithmic second derivative in (3.9) using
the identity

$\displaystyle\partial_{xy}\log
f=\frac{1}{f}\partial_{x}\partial_{y}f-(\partial_{x}\log f)(\partial_{y}\log
f)$ (3.10)

and relabeling the indices, we obtain

$\displaystyle\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma 1}}{\det
S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{s^{\alpha 1}}{\det
S}\right)\right)+\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\beta 1}}{\det
S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma 1}}{\det
S}\right)\right)$ $\displaystyle\qquad{}-\frac{\det S}{s^{\gamma
1}}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{s^{\gamma 1}}{\det
S}\right)=0,$

which is precisely (3.6). Finally, the relations (3.7) are obtained by setting
$\delta=\epsilon$ in the integrability condition (3.9), using the identity
(3.10), and the fact that the cofactors $s^{\gamma\alpha}$ and
$s^{\epsilon\alpha}$ are independent of the groups of variables ${\bf
x}^{\gamma}$ and ${\bf x}^{\epsilon}$ respectively. ∎

##  4 Different characterizations of Painlevé metrics

Let us start with the characterization of Painlevé metrics in terms of
complete additive separability of the Hamilton–Jacobi equations stated in
Proposition 1.2.

###### Proof of Proposition 1.2.

In block orthogonal coordinates $(\textbf{x}^{\alpha})$ associated to the
metric

$g=\sum_{\alpha=1}^{r}c_{\alpha}G_{\alpha},$

the Hamilton–Jacobi equation (1.3) reads

$\displaystyle\sum_{\alpha=1}^{r}c^{\alpha}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W}{\partial x^{i_{\alpha}}}\frac{\partial W}{\partial
x^{j_{\alpha}}}=E=:a_{1},$ (4.1)

where $c^{\alpha}=(c_{\alpha})^{-1}$. Assume that there exists a solution $W$
in the block-separable form (1.12) satisfying the completeness condition
(1.13). Then (4.1) can be written as

$\displaystyle\sum_{\alpha=1}^{r}c^{\alpha}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W_{\alpha}}{\partial x^{i_{\alpha}}}\frac{\partial W_{\alpha}}{\partial
x^{j_{\alpha}}}=E=:a_{1}.$ (4.2)

Differentiating (4.2) with respect to $a_{\beta}$, we get

$\displaystyle\sum_{\alpha=1}^{r}c^{\alpha}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}2\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W_{\alpha}}{\partial x^{i_{\alpha}}}\frac{\partial^{2}W_{\alpha}}{\partial
a_{\beta}\partial x^{j_{\alpha}}}=\delta_{1\beta}.$ (4.3)

From (1.12) and (1.13), it follows that the family of matrices

$\displaystyle
S(a_{1},\dots,a_{r})=(S_{\alpha\beta})(a_{1},\dots,a_{r}):=2\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W_{\alpha}}{\partial x^{i_{\alpha}}}\frac{\partial^{2}W_{\alpha}}{\partial
a_{\beta}\partial x^{j_{\alpha}}}$

are non-singular stäckel matrices of rank $r$ for all $(a_{1},\dots,a_{r})\in
U$. Using the invertibility of $S(a_{1},\dots,a_{r})$ which is equivalent to
the completeness condition (1.13), we get immediately from (4.3) that

$c^{\alpha}=\frac{s^{\alpha 1}}{\det S},$

which proves that $g$ is a Painlevé metric.

Conversely, assume that $g$ is a Painlevé metric of the form (1.10). Then the
Hamilton–Jacobi equation (1.3) takes the form

$\displaystyle\sum_{\alpha=1}^{r}\left(\frac{s^{\alpha 1}}{\det
S}\right)\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W}{\partial x^{i_{\alpha}}}\frac{\partial W}{\partial
x^{j_{\alpha}}}=E=:a_{1}.$ (4.4)

Now, since

$\sum_{\alpha=1}^{r}\left(\frac{s^{\alpha 1}}{\det
S}\right)s_{\alpha\beta}=\delta_{1\beta},$

we may rewrite (4.4) as

$\displaystyle\sum_{\alpha=1}^{r}\frac{s^{\alpha 1}}{\det
S}\left(\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W}{\partial x^{i_{\alpha}}}\frac{\partial W}{\partial
x^{j_{\alpha}}}-\sum_{\beta=1}^{r}s_{\alpha\beta}a_{\beta}\right)=0,$ (4.5)

for any $(a_{1},\dots,a_{r})\in U\subset\mathbb{R}^{r}$. Choosing $W$ in the
block-separable form (1.12), we see that any solution of the reduced
Hamilton–Jacobi equations

$\displaystyle\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W_{\alpha}}{\partial x^{i_{\alpha}}}\frac{\partial W_{\alpha}}{\partial
x^{j_{\alpha}}}=\sum_{\beta=1}^{r}s_{\alpha\beta}a_{\beta},$ (4.6)

will provide a solution of (4.5). But the latter equation always admit
_locally_ solutions $W_{\alpha}(\textbf{x}^{\alpha},\allowbreak
a_{1},\dots,a_{r})$ by standard PDE results [42]. Differentiating (4.6) with
respect to $a_{\beta}$, we obtain

$\displaystyle
2\sum_{\alpha=1}^{r}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\left(\frac{\partial
W_{\alpha}}{\partial
x^{i_{\alpha}}}\right)\left(\frac{\partial^{2}W_{\alpha}}{\partial
a_{\beta}\partial x^{j_{\alpha}}}\right)=s_{\alpha\gamma}.$

Since the generalized Stäckel matrix (1.8) is to be non-singular, it follows
that the rank condition that must be satisfied by our block-separable
parametrized family of solutions of the Hamilton–Jacobi equation is precisely
(1.13), thus proving that metrics of the Painlevé form (1.10) are indeed
characterized by the existence of a parametrized family of solutions of the
Hamilton–Jacobi equation satisfying a suitable completeness condition. ∎

Let us now give another proof of Proposition 1.2 that will allow us to
characterize Painlevé metrics in terms of the generalized Levi-Civita
conditions (3.6). Working in block-orthogonal coordinates or directly with a
Painlevé metric with the identification

$\displaystyle c^{\alpha}:=\frac{\det S}{s^{\alpha 1}},$ (4.7)

the Hamilton–Jacobi equation (1.3) takes the form

$\displaystyle\sum_{\alpha=1}^{r}c^{\alpha}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W}{\partial x^{i_{\alpha}}}\frac{\partial W}{\partial
x^{j_{\alpha}}}=E=:a_{1}.$ (4.8)

We recall that we seek a solution $W$ of (4.8) which is additively separable
into groups of variables, that is

$W=\sum_{\alpha=1}^{r}W_{\alpha}\big{(}{\bf x}^{\alpha}\big{)},$

and let

$u_{\alpha}^{(1)}=\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}j_{\alpha}}\frac{\partial
W_{\alpha}}{\partial x^{i_{\alpha}}}\frac{\partial W_{\alpha}}{\partial
x^{j_{\alpha}}},$

in which case the Hamilton–Jacobi equation (4.8) takes the form

$\sum_{\alpha=1}^{r}c^{\alpha}u_{\alpha}^{(1)}=a_{1}.$

Differentiating the latter equation with respect to $x^{k_{\beta}}$, we obtain
the first order differential system in normal form

$\displaystyle\partial_{k_{\beta}}u^{(1)}_{\gamma}=\begin{cases}\displaystyle-\frac{1}{c^{\beta}}\sum_{\alpha=1}^{r}\big{(}\partial_{k_{\beta}}c^{\alpha}\big{)}u^{(1)}_{\alpha}&\gamma=\beta,\\\
0&\gamma\neq\beta.\end{cases}$ (4.9)

Introducing the first-order differential operators

$\displaystyle D_{k_{\beta}}:=\frac{\partial}{\partial
x^{k_{\beta}}}-\frac{1}{c^{\beta}}\sum_{\alpha=1}^{r}\big{(}\partial_{k_{\beta}}c^{\alpha}\big{)}u^{(1)}_{\alpha}\frac{\partial}{\partial
u^{(1)}_{\beta}},$ (4.10)

the differential system (4.9) will admit a family of solutions
$u^{(1)}_{\alpha}=u^{(1)}_{\alpha}\big{(}x^{1},\dots,x^{n};a_{1},\dots,a_{r}\big{)}$
defined on an open subset $U\in M$ and depending smoothly on $r$ constants
$(a_{1},\dots,a_{r})$ defined in an open subset $A\subset{\mathbb{R}}^{r}$,
satisfying the rank condition

$\displaystyle\det\left(\frac{\partial u^{(1)}_{\alpha}}{\partial
a_{\beta}}\right)\neq 0,$ (4.11)

if and only if the operators (4.10) pairwise commute, that is

$\displaystyle\big{[}D_{k_{\beta}},D_{j_{\alpha}}\big{]}=0$ (4.12)

for all $1\leq\alpha,\beta\leq r$, $1_{\alpha}\leq j_{\alpha}\leq l_{\alpha}$,
$1_{\beta}\leq k_{\beta}\leq l_{\beta}$. We refer to [2, Theorem 2.1] for this
result. Note that if (4.12) hold, then the Hamilton–Jacobi equation admits
locally a solution which is additively separable into groups of variables and
satisfies the completeness condition (1.13) as a consequence of the
completeness condition (4.11). In consequence, such metrics are of the
Painlevé form (1.10).

We now prove that the commutation relations (4.12) are equivalent to the
generalized Levi-Civita conditions (3.6). Note that this is a natural
generalization of the link between the complete separation of variables which
is familiar from the Stäckel case and the classical Levi-Civita separability
conditions, as reviewed in [2, 27]. Indeed, we have

######  Lemma 4.1.

The pairwise commutation relations (4.12) for the derivations $D_{k_{\beta}}$
are equivalent to the generalized Levi-Civita conditions (3.6).

###### Proof.

Substituting the expression (4.10) of the differential operators
$D_{k_{\beta}}$ into the commutation relations (4.12), we obtain

$\displaystyle\left[-\partial_{k_{\beta}}\left(\frac{1}{c^{\gamma}}\sum_{\delta=1}^{r}\big{(}\partial_{p_{\gamma}}c^{\delta}\big{)}u^{(1)}_{\delta}\right)+\frac{1}{c^{\beta}c^{\gamma}}\sum_{\alpha=1}^{r}\big{(}\partial_{k_{\beta}}c^{\alpha}\big{)}u^{(1)}_{\alpha}\big{(}\partial_{p_{\gamma}}c^{\beta}\big{)}\right]\frac{\partial}{\partial
u^{(1)}_{\gamma}}$
$\displaystyle\qquad{}+\left[-\partial_{k_{\beta}}\left(\frac{1}{c^{\gamma}}\sum_{\delta=1}^{r}\big{(}\partial_{p_{\gamma}}c^{\delta}\big{)}u^{(1)}_{\delta}\right)+\frac{1}{c^{\beta}c^{\gamma}}\sum_{\alpha=1}^{r}\big{(}\partial_{k_{\beta}}c^{\alpha}\big{)}u^{(1)}_{\alpha}\big{(}\partial_{p_{\gamma}}c^{\beta}\big{)}\right]\frac{\partial}{\partial
u^{(1)}_{\gamma}}=0.$

For $1\leq\gamma\neq\beta\leq r$, the above identity is equivalent to

$\displaystyle\big{(}\partial_{k_{\beta}}\log
c^{\gamma}\big{)}\big{(}\partial_{p_{\gamma}}\log
c^{\alpha}\big{)}+\big{(}\partial_{k_{\beta}}\log
c^{\alpha}\big{)}\big{(}\partial_{p_{\gamma}}\log
c^{\beta}\big{)}-\frac{1}{c^{\alpha}}\partial_{k_{\beta}}\partial_{p_{\gamma}}c^{\alpha}=0,$
(4.13)

which are precisely the generalized Levi-Civita conditions (3.6) after
relabeling of indices. Note that for $1\leq\gamma=\beta\leq r$, the above
identity is always satisfied by a straightforward calculation. ∎

At this stage, we thus have proved another characterization of Painlevé
metrics which appears in [10, p. 12].

######  Proposition 4.2.

A metric $g$ is of Painlevé type if and only if there exist block orthogonal
coordinates $\big{(}\textbf{x}^{\alpha}\big{)}$ such that the generalized
Levi-Civita conditions (4.13) hold.

We finish this section giving still another characterization of Painlevé
metrics of a more intrinsic nature. The starting point is the observation that
the generalized Killing–Eisenhart equations are related to the existence of
quadratic first integrals (or symmetries) $K_{(\beta)}$ by the following
result proved in [10, Proposition 5.3] (see also Lemma 3.1 for an implicit
proof of this proposition)

######  Proposition 4.3.

In block orthogonal coordinates, we have that

$\big{\\{}H,K_{(\beta)}\big{\\}}=0,$

if and only if the Killing–Eisenhart equations

$\displaystyle\partial_{j_{\gamma}}(\rho_{\beta\delta})=(\rho_{\beta\gamma}-\rho_{\beta\delta})\big{(}\partial_{j_{\gamma}}\log
c^{\delta}\big{)},$ (4.14)

hold for all $1\leq\gamma,\delta\leq r$.

The second observation is the fact that the integrability conditions for the
Killing–Eisenhart equations (4.14) are given by

$\displaystyle(\rho_{\beta\gamma}-\rho_{\beta\delta})\left[\big{(}\partial_{j_{\gamma}}\log
c^{\delta}\big{)}\big{(}\partial_{k_{\alpha}}\log
c^{\gamma}\big{)}+\big{(}\partial_{j_{\gamma}}\log
c^{\alpha}\big{)}\big{(}\partial_{k_{\alpha}}\log
c^{\delta}\big{)}-\frac{1}{c^{\delta}}\partial_{k_{\alpha}}\partial_{j_{\gamma}}c^{\delta}\right]=0,$

for all $1\leq\alpha,\beta,\gamma,\delta\leq r$. Clearly, these integrability
conditions can be shortened as

$\displaystyle(\rho_{\beta\gamma}-\rho_{\beta\delta})\cdot\left[\textrm{Levi-
Civita conditions}\right]=0.$ (4.15)

Using these two observations, Chanu ad Rastelli proved in [10, Proposition
5.5] the following characterization of Painlevé metrics.

######  Proposition 4.4.

$(M,g)$ is a Painlevé manifold if and only if

  1. $1)$

there exist $r$ independent quadratic first integral $K_{(\beta)}$,
$\beta=1,\dots,r$ such that $K_{(1)}=H$ and $\\{H,K_{(\beta)}\\}=0$.

  2. $2)$

The associated Killing two tensors $K_{(\beta)}^{ij}$ are simultaneously
block-diagonalized and have common normally integrable eigenspaces.

###### Proof.

If $g$ is a Painlevé metric, then the above assertions were already proved.

Assume now that there exist $r$ linearly independent Killing tensors
simultaneously in block diagonal form. Then the generalized Killing–Eisenhart
equations (4.14) will admit an $r$-dimensional vector space of solutions. The
latter is equivalent to the invertibility of the fundamental matrix
$\mathcal{A}$ defined by

${\mathcal{A}}=\left(\begin{matrix}\rho_{11}&\dots&\rho_{1r}\\\
\vdots&&\vdots\\\ \rho_{r1}&\dots&\rho_{rr}\end{matrix}\right).$

Hence any solution $(\rho_{1},\dots,\rho_{r})$ of (4.14) is given by

$\left(\begin{matrix}\rho_{1}\\\ \vdots\\\
\rho_{r}\end{matrix}\right)={\mathcal{A}}\left(\begin{matrix}a_{1}\\\
\vdots\\\ a_{r}\end{matrix}\right),$

for some constants $(a_{1},\dots,a_{r})\in A$. It is clear then that the
Killing–Eisenhart equations are completely integrable and thus satisfy the
integrability conditions (4.15). Moreover, from the invertibility of
$\mathcal{A}$, we can always choose the constants $(a_{1},\dots,a_{r})$ such
that $\rho_{\alpha}\neq\rho_{\beta}$, $\forall\,1\leq\alpha\neq\beta\leq r$ at
a point $p\in M$ and therefore in an neighbourhood of $p$, by continuity.
Hence, the integrability conditions (4.15) reduces to the generalized Levi-
Civita separability conditions (4.13). We conclude that $g$ is a Painlevé
metric from Proposition 4.2. ∎

As a concluding remark for this section, we emphasize that the hypotheses
(3.5) that we make to deduce the Levi-Civita conditions from the
Killing–Eisenhart equations aren’t in fact necessary. Indeed, it follows from
Proposition 4.2 or Proposition 4.4 that the Levi-Civita conditions always hold
whenever $g$ is a Painlevé metric, that is a metric of the form (1.10).

##  5 The generalized Robertson conditions and the separability
of the Helmholtz equation

Assume that the manifold $(M,g)$ admits locally block-orthogonal coordinates
$\big{(}\textbf{x}^{\alpha}\big{)}$ such that
$g=\sum\limits_{\beta=1}^{r}c_{\beta}G_{\beta}$. Then using the same
calculation that led to the expression (1.19) of the Laplace–Beltrami
operator, the Helmholtz equation (1.25) for such a metric reads

$\displaystyle\sum_{\beta=1}^{r}c^{\beta}\left[-\Delta_{G_{\beta}}+\sum_{j_{\beta}=1_{\beta}}^{l_{\beta}}\gamma^{j_{\beta}}\partial_{j_{\beta}}\right]u=a_{1}u,$
(5.1)

where $c^{\beta}=(c_{\beta})^{-1}$ and

$\gamma^{j_{\beta}}=-\sum_{i_{\beta}=1_{\beta}}^{l_{\beta}}\big{(}G^{\beta}\big{)}^{i_{\beta}j_{\beta}}\partial_{i_{\beta}}\left[\log\left(\frac{c_{1}^{\frac{l_{1}}{2}}\cdots
c_{r}^{\frac{l_{r}}{2}}}{c_{\beta}}\right)\right].$

We shall say that a block diagonal metric
$g=\sum\limits_{\beta=1}^{r}c_{\beta}G_{\beta}$ satisfies the generalized
Robertson conditions if and only if the differential equations

$\displaystyle\partial_{i_{\alpha}}\gamma^{j_{\beta}}=0,\qquad\forall\,1\leq\alpha\neq\beta\leq
r,$ (5.2)

hold.

Note that under the assumption (5.2), we may write the Helmholtz equation
(5.1) as

$\sum_{\beta=1}^{r}c^{\beta}B_{\beta}u=a_{1}u,$

where the partial differential operators $B_{\beta}$, $1\leq\beta\leq r,$
defined by

$B_{\beta}:=-\Delta_{G_{\beta}}+\sum_{j_{\beta}=1_{\beta}}^{l_{\beta}}\gamma^{j_{\beta}}\partial_{j_{\beta}},$

now depend on the group of variables ${\bf x}^{\beta}$ only.

In this section, we want to prove

######  Proposition 5.1.

Assume that the manifold $(M,g)$ admits locally block-orthogonal coordinates
and satisfies the Robertson conditions (5.2). Then $g$ is a Painlevé metric if
and only if there exists a parametrized family of solutions $u$ of the
Helmholtz equation (5.1) which is product-separable into groups of variables,
of the form

$\displaystyle u=\prod_{\beta=1}^{r}u_{\beta}\big{(}{\bf
x}^{\beta};a_{1},\dots,a_{r}\big{)},$ (5.3)

and satisfies the completeness condition

$\displaystyle\det\left({\partial_{a_{\alpha}}\left(\frac{B_{\beta}u_{\beta}}{u_{\beta}}\right)}\right)\neq
0.$ (5.4)

###### Proof.

Assume that $g$ is a Painlevé metric. This means that $c_{\beta}=\frac{\det
S}{s^{\beta 1}}$. Then the Helmholtz equation (5.1) may equivalently be
written in terms of the generalized Stäckel matrix $S=(s_{\alpha\beta})$ and a
set of arbitrary real parameters $(a_{1}:=\lambda,a_{2},\dots,a_{r})$ defined
on an open subset of $A$ of ${\mathbb{R}}^{r}$ as

$\displaystyle\sum_{\beta=1}^{r}c^{\beta}\left[B_{\beta}-\sum_{\alpha=1}^{r}s_{\beta\alpha}a_{\alpha}\right]u=0.$
(5.5)

We now consider a parametrized family of solutions $u$ of the Helmholtz
equation (5.5) which is product-separable into groups of variables, of the
form (5.3) where for each $1\leq\beta\leq r$, the factor $u_{\beta}$ is
required to satisfy the partial differential equation in the group of
variables ${\bf x}^{\beta}$ given by

$\displaystyle
B_{\beta}u_{\beta}=\left(\sum_{\alpha=1}^{r}s_{\beta\alpha}a_{\alpha}\right)u_{\beta}.$
(5.6)

We note that for Painlevé metrics of Riemannian signature on a compact
manifold, it follows by [20, Theorem 8.3] that the elliptic partial
differential equation (5.6) admits a unique solution if the parameters
$(a_{1}:=\lambda,a_{2},\dots,a_{r})$ are chosen so that the non-positivity
condition

$\sum_{\alpha=1}^{r}s_{\beta\alpha}a_{\alpha}\leq 0,$

is satisfied (at least locally).

The form of the separated equations (5.6) and the assumption that the
generalized Stäckel matrix is invertible imply that our parametrized family of
block-separable solutions of the Helmholtz equation must satisfy the rank
condition (5.4), i.e.,

$\det\left({\partial_{a^{\alpha}}\left(\frac{B_{\beta}u_{\beta}}{u_{\beta}}\right)}\right)\neq
0.$

We now prove the converse statement, namely that the existence of a
parametrized family of block-separable solutions (5.3) of the Helmholtz
equation satisfying partial differential equations of the form

$\displaystyle\sum_{\beta=1}^{r}c^{\beta}B_{\beta}u=a_{1}u,$ (5.7)

and the rank condition (5.4) implies that the underlying metric must be of
Painlevé form. Substituting $u$ of the form (5.3) into (5.7) gives

$\sum_{\beta=1}^{r}c^{\beta}\frac{B_{\beta}u_{\beta}}{u_{\beta}}=a_{1}.$

Differentiating the latter equation with respect to $a_{\alpha}$, we obtain

$\sum_{\beta=1}^{r}c^{\beta}\left({\partial_{a^{\alpha}}\left(\frac{B_{\beta}u_{\beta}}{u_{\beta}}\right)}\right)=\delta^{\alpha}_{1}.$

Letting

$s_{\beta\alpha}={\partial_{a^{\alpha}}\left(\frac{B_{\beta}u_{\beta}}{u_{\beta}}\right)},$

we obtain the expression of $c_{\beta}$ for a Painlevé metric as given in
(4.7). ∎

######  Remark 5.2.

From (5.6) and the fact that the Stäckel matrix $S$ is invertible, we conclude
that the product separable solutions (1.26) satisfy eigenvalue equations of
the form

$\displaystyle T_{\alpha}u=a_{\alpha}u,\qquad 1\leq\alpha\leq r,$ (5.8)

where the $T_{\alpha}$ are the linear second order differential operators
given by

$T_{\alpha}=\sum_{\beta=1}^{r}\frac{s^{\beta\alpha}}{\det S}B_{\beta}.$

They will be shown in Theorem 1.5 to be identical to the operators
$\Delta_{K_{(\alpha)}}$ defined by (1.28). Hence the separation constants
$a_{1},\dots,a_{r}$ can be understood as the natural eigenvalues of the
operators $\Delta_{K_{(\alpha)}}$.

##  6 Proofs of the main theorems

###  6.1 Proof of Theorem 1.4

The fact that the generalized Robertson conditions (1.20) are sufficient
conditions for the product separability of the Helmholtz equation (1.25) in
the groups of variables associated to a Painlevé metric follows from
Proposition 5.1.

What remains to be done in order to prove Theorem 1.4 and what constitutes our
main task is therefore to show that the generalized Robertson conditions are
equivalent to the conditions (1.27) on the Ricci tensor, thus generalizing the
classical result of Eisenhart [18] to Painlevé metrics. In order to do this,
we will show that

$\displaystyle
R_{j_{\alpha}k_{\beta}}=-\frac{3}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right]+\frac{1}{4}T_{j_{\alpha}k_{\beta}},$
(6.1)

where

$\displaystyle T_{j_{\alpha}k_{\beta}}=(l_{\alpha}+l_{\beta}-2)\frac{s^{\alpha
1}s^{\beta 1}}{\det
S}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{\det S}{s^{\alpha
1}s^{\beta 1}}\right)$
$\displaystyle\hphantom{T_{j_{\alpha}k_{\beta}}=}{}+\sum_{\gamma\neq\alpha,\beta=1}^{r}l_{\gamma}\left[\left(\partial_{j_{\alpha}}\log\frac{s^{\gamma
1}}{\det S}\right)\left(\partial_{k_{\beta}}\log\frac{s^{\alpha 1}}{\det
S}\right)\right.$
$\displaystyle\left.\hphantom{T_{j_{\alpha}k_{\beta}}=}{}+\left(\partial_{j_{\alpha}}\log\frac{s^{\beta
1}}{\det S}\right)\left(\partial_{k_{\beta}}\log\frac{s^{\gamma 1}}{\det
S}\right)-\frac{\det S}{s^{\gamma
1}}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{s^{\gamma 1}}{\det
S}\right)\right].$ (6.2)

We note that the expression (6.1) of the off block-diagonal components of the
Ricci tensor $R_{j_{\alpha}k_{\beta}}$ is independent of the $r$ Riemannian
block metrics $G_{\beta}$, $1\leq\beta\leq r$ defined by (1.11). We also
remark that the first term in the expression (6.2) of
$T_{j_{\alpha}k_{\beta}}$, involving second derivatives, vanishes identically
in the special case of Stäckel metrics since the pre-factor
$l_{\alpha}+l_{\beta}-2$ is zero in that case.

Once we will have established (6.1), it will then follow from Lemma 3.2 and
more precisely from the generalized Levi-Civita conditions (3.6) and (3.7)
that the generalized Robertson conditions (1.20) are indeed equivalent to the
vanishing conditions (1.27) on the non-block diagonal components of the Ricci
tensor. We therefore proceed to establish the form (6.1) of the Ricci tensor
for a Painlevé metric.

The expression of the Ricci tensor in terms of the Christoffel symbols is
given by

$\displaystyle
R_{j_{\alpha}k_{\beta}}=R^{l}_{{\phantom{l}}lj_{\alpha}k_{\beta}}=\partial_{l}{\Gamma^{l}}_{j_{\alpha}k_{\beta}}-\partial_{j_{\alpha}}{\Gamma^{l}}_{lk_{\beta}}+{\Gamma^{m}}_{j_{\alpha}k_{\beta}}{\Gamma^{l}}_{lm}-{\Gamma^{m}}_{lk_{\beta}}{\Gamma^{l}}_{j_{\alpha}m},$
(6.3)

where the summation convention is applied with $1\leq l,m\leq n=\dim M$. In
order to compute the right-hand side of (6.3), we will need expressions for
the Christoffel symbols of a Painlevé metric (1.10). Using the standard
formulas

$\displaystyle\Gamma_{hji}=\frac{1}{2}(\partial_{h}g_{ji}+\partial_{j}g_{ih}-\partial_{i}g_{hj}),\qquad{\Gamma^{i}}_{hj}=g^{ik}\Gamma_{hjk},$

and writing the Painlevé metric (1.10) in block-diagonal form as

${\rm
d}s^{2}=\sum_{\alpha=1}^{r}\sum_{i_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{j_{\alpha}=1_{\alpha}}^{l_{\alpha}}(g_{\alpha})_{i_{\alpha}j_{\alpha}}{\rm
d}x^{i_{\alpha}}{\rm d}x^{j_{\alpha}},$

we obtain for fixed indices $1\leq\alpha,\beta\leq r$,

$\displaystyle{\Gamma^{i_{\alpha}}}_{k_{\alpha}j_{\beta}}=\frac{1}{2}\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}g^{\alpha}\big{)}^{i_{\alpha}p_{\alpha}}\partial_{j_{\beta}}(g_{\alpha})_{k_{\alpha}p_{\alpha}},$
$\displaystyle{\Gamma^{j_{\beta}}}_{i_{\alpha}k_{\alpha}}=-\frac{1}{2}\sum_{p_{\beta}=1_{\beta}}^{l_{\beta}}\big{(}g^{\beta}\big{)}^{j_{\beta}p_{\beta}}\partial_{p_{\beta}}(g_{\alpha})_{i_{\alpha}k_{\alpha}}\qquad\text{for}\quad\alpha\neq\beta,$
(6.4)

and

$\displaystyle{\Gamma^{i_{\alpha}}}_{h_{\alpha}k_{\alpha}}=\frac{1}{2}\sum_{k_{\alpha}=1_{\alpha}}^{l_{\alpha}}g^{i_{\alpha}k_{\alpha}}(\partial_{h_{\alpha}}g_{j_{\alpha}k_{\alpha}}+\partial_{j_{\alpha}}g_{h_{\alpha}k_{\alpha}}-\partial_{k_{\alpha}}g_{h_{\alpha}j_{\alpha}}).$
(6.5)

In view of the expressions (6.4) of the Christoffel symbols, it is convenient
to split the sum over $l$ appearing in (6.3) into three sums, the first sum
corresponding to the values of the summation index $l$ lying in groups of
indices different from the groups corresponding to $\alpha$ and $\beta$, and
the remaining two to the values of $l$ belonging to the groups of indices
labelled by $\alpha$ and $\beta$ respectively. Thus we write

$\displaystyle
R_{j_{\alpha}k_{\beta}}=\sum_{l=1}^{n}R^{l}_{{\phantom{l}}lj_{\alpha}k_{\beta}}=\sum_{\gamma\neq\alpha,\,\beta=1}^{r}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}+\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}+\sum_{p_{\beta}=1_{\beta}}^{l_{\beta}}R^{p_{\beta}}_{{\phantom{l}}p_{\beta}j_{\alpha}k_{\beta}}.$
(6.6)

Let us begin with the first term. We have, for $\gamma\neq\alpha,\beta$,

$\displaystyle\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=-\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\partial_{j_{\alpha}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}k_{\beta}}+\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}m_{\alpha}}$
$\displaystyle\hphantom{\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=}{}+\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{m_{\beta}=1_{\beta}}^{l_{\beta}}{\Gamma^{m_{\beta}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}m_{\beta}}-\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{m_{\gamma}=1_{\gamma}}^{l_{\gamma}}{\Gamma^{m_{\gamma}}}_{p_{\gamma}k_{\beta}}{\Gamma^{p_{\gamma}}}_{j_{\alpha}m_{\gamma}},$
(6.7)

where the summations have been written out explicitly to avoid notational
ambiguities. We begin with evaluating the first-derivative term on the right-
hand side of (6.7) using the expressions (6.4) for the Christoffel symbols,
thus obtaining, for $\gamma\neq\alpha,\beta$,

$\displaystyle\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\partial_{j_{\alpha}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}k_{\beta}}=\frac{1}{2}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{n_{\gamma}=1_{\gamma}}^{l_{\gamma}}\partial_{j_{\alpha}}\big{(}\big{(}g^{\gamma}\big{)}^{p_{\gamma}n_{\gamma}}\partial_{k_{\beta}}((g_{\gamma})_{p_{\gamma}n_{\gamma}})\big{)}=\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log(|g_{\gamma}|),$
(6.8)

where $|g_{\gamma}|:=\det(g_{i_{\gamma}j_{\gamma}})$. It follows that

$\sum_{\gamma\neq\alpha,\,\beta=1}^{r}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\partial_{j_{\alpha}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}k_{\beta}}=\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\prod_{\gamma\neq\alpha,\,\beta=1}^{r}\log(|g_{\gamma}|)\right).$

From the Painlevé form (1.10), we have

$\displaystyle g_{i_{\gamma}j_{\gamma}}=\frac{\det S}{s^{\gamma
1}}(G_{\gamma})_{i_{\gamma}j_{\gamma}},$ (6.9)

so that

$|g_{\gamma}|=\frac{(\det S)^{l_{\gamma}}}{\big{(}{s^{\gamma
1}}\big{)}^{l_{\gamma}}}|G_{\gamma}|.$

Using the fact that $|G_{\gamma}|$ is a function of the variables ${\bf
x}^{\gamma}$ only, we obtain

$\displaystyle\sum_{\gamma\neq\alpha,\,\beta=1}^{r}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\partial_{j_{\alpha}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}k_{\beta}}=\frac{1}{2}\sum_{\gamma\neq\alpha,\,\beta=1}^{r}l_{\gamma}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\gamma 1}}\right)\right).$

We notice that the above expression is independent of the quadratic
differential forms $G_{\alpha}$ defined by (1.11).

Next we evaluate the terms quadratic in the Christoffel symbols in the right-
hand side of (6.7). Again, we use the fact that $\gamma\neq\alpha,\beta$ and
the fact that in the Painlevé form (1.10), each of the quadratic differential
forms $G_{\alpha}$ defined by (1.11) depends on the group of variables ${\bf
x}^{\alpha}$ only. We have

$\displaystyle\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}m_{\alpha}}$
$\displaystyle\qquad{}=\frac{1}{4}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{n_{\alpha}=1_{\alpha}}^{l_{\alpha}}(g^{\alpha})^{m_{\alpha}n_{\alpha}}\partial_{k_{\beta}}((g_{\alpha})_{j_{\alpha}n_{\alpha}})g^{p_{\gamma}h_{\gamma}}\partial_{m_{\alpha}}((g_{\gamma})_{p_{\gamma}h_{\gamma}})$
$\displaystyle\qquad{}=\frac{1}{4}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{n_{\alpha}=1_{\alpha}}^{l_{\alpha}}(g^{\alpha})^{m_{\alpha}n_{\alpha}}\partial_{k_{\beta}}((g_{\alpha})_{j_{\alpha}n_{\alpha}})\partial_{m_{\alpha}}\log(|g_{\gamma}|)$
$\displaystyle\qquad{}=\frac{1}{4}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{n_{\alpha}=1_{\alpha}}^{l_{\alpha}}\frac{s^{\alpha
1}}{\det
S}\big{(}G^{\alpha}\big{)}^{m_{\alpha}n_{\alpha}}\partial_{k_{\beta}}\left(\frac{\det
S}{s^{\alpha
1}}(G_{\alpha})_{j_{\alpha}n_{\alpha}}\right)\partial_{m_{\alpha}}\log(|g_{\gamma}|)$
$\displaystyle\qquad{}=\frac{1}{4}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{n_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\delta^{m_{\alpha}}}_{j_{\alpha}}\partial_{k_{\beta}}\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\partial_{m_{\alpha}}\log\left(\frac{(\det
S)^{l_{\gamma}}}{(s^{\gamma 1})^{l_{\gamma}}}\right)$
$\displaystyle\qquad{}=\frac{1}{4}\left(\partial_{k_{\beta}}\log\left(\frac{s^{\alpha
1}}{\det
S}\right)\right)\left(\partial_{j_{\alpha}}\log\left(\frac{\big{(}s^{\gamma
1}\big{)}^{l_{\gamma}}}{(\det S)^{l_{\gamma}}}\right)\right).$ (6.10)

We notice that the above expression is again independent of the quadratic
differential forms $G_{\alpha}$ defined by (1.11). Likewise, we obtain for the
next quadratic term in the Christoffel symbols that appears in the right-hand
side of (6.7),

$\displaystyle\sum_{m_{\beta}=1_{\beta}}^{l_{\beta}}\sum_{p_{\gamma}=1_{\gamma}}^{l_{\gamma}}{\Gamma^{m_{\beta}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\gamma}}}_{p_{\gamma}m_{\beta}}=\frac{1}{4}\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\beta
1}}{\det
S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{\big{(}s^{\gamma
1}\big{)}^{p_{\gamma}}}{(\det S)^{p_{\gamma}}}\right)\right).$ (6.11)

For the third and final quadratic term, we have

$\displaystyle\sum_{m_{\gamma}=1_{\gamma}}^{l_{\gamma}}{\Gamma^{m_{\gamma}}}_{p_{\gamma}k_{\beta}}{\Gamma^{p_{\gamma}}}_{j_{\alpha}m_{\gamma}}$
$\displaystyle=\frac{1}{4}\sum_{m_{\gamma}=1_{\gamma}}^{l_{\gamma}}{\delta^{m_{\gamma}}}_{p_{\gamma}}\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma
1}}{\det
S}\right)\right){\delta^{p_{\gamma}}}_{m_{\gamma}}\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)$
$\displaystyle=\frac{1}{4}l_{\gamma}\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right),$ (6.12)

which is likewise independent of the quadratic differential forms $G_{\alpha}$
defined by (1.11). Putting together the expressions (6.8), (6.10), (6.11) and
(6.12) we obtain

$\displaystyle\sum_{1\leq\gamma\neq\alpha,\beta\leq
r}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=\sum_{\gamma\neq\alpha,\,\beta=1}^{r}l_{\gamma}\left[\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right.$
$\displaystyle\hphantom{\sum_{1\leq\gamma\neq\alpha,\beta\leq
r}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}\left(\partial_{k_{\beta}}\log\left(\frac{s^{\alpha
1}}{\det S}\right)\right)\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)$
$\displaystyle\hphantom{\sum_{1\leq\gamma\neq\alpha,\beta\leq
r}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\beta
1}}{\det S}\right)\right)\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)$
$\displaystyle\left.\hphantom{\sum_{1\leq\gamma\neq\alpha,\beta\leq
r}R^{p_{\gamma}}_{{\phantom{l}}p_{\gamma}j_{\alpha}k_{\beta}}=}{}-\frac{1}{4}\left(\partial_{k_{\beta}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)\left(\partial_{j_{\alpha}}\log\left(\frac{s^{\gamma
1}}{\det S}\right)\right)\right].$ (6.13)

We still need to evaluate the curvature components
$R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}$ and
$R^{p_{\beta}}_{{\phantom{l}}p_{\beta}j_{\alpha}k_{\beta}}$, which will
require a separate calculation. We have

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}=\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\partial_{p_{\alpha}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}k_{\beta}}-\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\partial_{j_{\alpha}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}k_{\beta}}$
$\displaystyle\hphantom{\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}=}{}+\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}m_{\alpha}}+\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\beta}=1_{\beta}}^{l_{\beta}}{\Gamma^{m_{\beta}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}m_{\beta}}$
$\displaystyle\hphantom{\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}=}{}-\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{p_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}m_{\alpha}}-\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\beta}=1_{\beta}}^{l_{\beta}}{\Gamma^{m_{\beta}}}_{l_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}m_{\beta}},$
(6.14)

where again we have written out the summation signs explicitly to avoid
notational ambiguities. We have, using (6.4),

$\displaystyle{\Gamma^{p_{\alpha}}}_{j_{\alpha}k_{\beta}}=\frac{1}{2}{\delta^{p_{\alpha}}}_{j_{\alpha}}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right),$ (6.15)

so using the fact that the cofactor $s^{\alpha 1}$ is independent of ${\bf
x}^{\alpha}$, we obtain

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\partial_{p_{\alpha}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}k_{\beta}}=\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log(\det
S)\big{)},$ (6.16)

and a similar calculation gives

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\partial_{j_{\alpha}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}k_{\beta}}=\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log\big{(}(\det
S)^{l_{\alpha}}\big{)}\big{)}.$ (6.17)

We now evaluate the quadratic terms in the Christoffel symbols that appear in
the curvature component (6.14). In order to do so, we substitute into the
expression (6.5) of the Christoffel symbols the expressions

$(g_{\alpha})_{i_{\alpha}j_{\alpha}}=\frac{\det S}{s^{\alpha 1}}G_{\alpha},$

which result from the Painlevé form (1.10). We obtain the following
expressions for the Christoffel symbols,

$\displaystyle{\Gamma^{i_{\alpha}}}_{h_{\alpha}j_{\alpha}}={\gamma^{i_{\alpha}}}_{h_{\alpha}j_{\alpha}}+\frac{1}{2}\Bigg{[}{\delta^{i_{\alpha}}}_{j_{\alpha}}\partial_{h_{\alpha}}\big{(}\log{(\det
S)}\big{)}+{\delta^{i_{\alpha}}}_{h_{\alpha}}\partial_{j_{\alpha}}\big{(}\log{(\det
S)}\big{)}$
$\displaystyle\hphantom{{\Gamma^{i_{\alpha}}}_{h_{\alpha}j_{\alpha}}=}{}-\sum_{k_{\alpha}=1_{\alpha}}^{l_{\alpha}}\big{(}G^{\alpha}\big{)}^{i_{\alpha}k_{\alpha}}(G_{\alpha})_{h_{\alpha}j_{\alpha}}\partial_{k_{\alpha}}\big{(}\log{(\det
S)}\big{)}\Bigg{]},$ (6.18)

where the ${\gamma^{i_{\alpha}}}_{h_{\alpha}j_{\alpha}}$ denote the
Christoffel symbols of the block metric $G_{\alpha}$ given by (1.11). It then
follows from (6.18) and (6.15) that

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}m_{\alpha}}$
$\displaystyle\qquad{}=\frac{1}{2}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha
1}}\right)\right)\left[\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\gamma^{p_{\alpha}}}_{p_{\alpha}j_{\alpha}}+\frac{1}{2}l_{\alpha}\partial_{m_{\alpha}}\big{(}\log{(\det
S)}\big{)}\right].$ (6.19)

Similarly, using (6.15) and (6.18), we obtain

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{p_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}m_{\alpha}}$
$\displaystyle\qquad{}=\frac{1}{2}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha
1}}\right)\right)\left[\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\gamma^{p_{\alpha}}}_{j_{\alpha}p_{\alpha}}+\frac{1}{2}l_{\alpha}\partial_{m_{\alpha}}\big{(}\log{(\det
S)}\big{)}\right].$ (6.20)

It follows therefore from (6.19) and (6.20) that

$\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}m_{\alpha}}-\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{p_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}m_{\alpha}}=0.$

We now evaluate the remaining difference of two double sums in the expression
(6.14) of the curvature, using the expressions

${\Gamma^{m_{\beta}}}_{j_{\alpha}k_{\beta}}=\frac{1}{2}{\delta^{m_{\beta}}}_{k_{\beta}}\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta
1}}\right)\right),\qquad{\Gamma^{m_{\beta}}}_{j_{\alpha}k_{\beta}}=\frac{1}{2}\partial_{m_{\beta}}\left(\log\left(\frac{(\det
S)^{l_{\alpha}}}{\big{(}s^{\beta 1}\big{)}^{l_{\alpha}}}\right)\right),$

for the Christoffel symbols, to obtain

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\alpha}=1_{\alpha}}^{l_{\alpha}}{\Gamma^{m_{\alpha}}}_{j_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{p_{\alpha}m_{\alpha}}-\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}\sum_{m_{\beta}=1_{\beta}}^{l_{\beta}}{\Gamma^{m_{\beta}}}_{l_{\alpha}k_{\beta}}{\Gamma^{p_{\alpha}}}_{j_{\alpha}m_{\beta}}$
$\displaystyle\qquad{}=\frac{1}{4}(l_{\alpha}-1)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta 1}}\right)\right)\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right).$ (6.21)

Substituting the expressions (6.16), (6.17), (6.19), (6.20) and (6.21) into
(6.14), we obtain

$\displaystyle\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}=\frac{1-l_{\alpha}}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log(\det
S)\big{)}$
$\displaystyle\hphantom{\sum_{p_{\alpha}=1_{\alpha}}^{l_{\alpha}}R^{p_{\alpha}}_{{\phantom{l}}p_{\alpha}j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}(l_{\alpha}-1)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta 1}}\right)\right)\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right),$ (6.22)

and similarly

$\displaystyle\sum_{p_{\beta}=1_{\beta}}^{l_{\alpha}}R^{p_{\beta}}_{{\phantom{l}}p_{\beta}j_{\alpha}k_{\beta}}=\frac{1-l_{\beta}}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log(\det
S)\big{)}$
$\displaystyle\hphantom{\sum_{p_{\beta}=1_{\beta}}^{l_{\alpha}}R^{p_{\beta}}_{{\phantom{l}}p_{\beta}j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}(l_{\beta}-1)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta 1}}\right)\right)\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right).$ (6.23)

We now substitute the expressions (6.13), (6.22) and (6.1) into the
decomposition (6.6) of the off-block diagonal Ricci curvature components
$R_{j_{\alpha}k_{\beta}}$ to obtain, for $\alpha\neq\beta$,

$\displaystyle
R_{j_{\alpha}k_{\beta}}=-\frac{1}{2}\sum_{\gamma=1}^{r}l_{\gamma}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\gamma
1}}\right)\right)+\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log(\det
S)\big{)}$
$\displaystyle\hphantom{R_{j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}(l_{\alpha}+l_{\beta}-2)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta 1}}\right)\right)\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right)$
$\displaystyle\hphantom{R_{j_{\alpha}k_{\beta}}=}{}+\frac{1}{4}\sum_{\gamma\neq\alpha,\,\beta=1}^{r}l_{\gamma}\left[\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\alpha 1}}\right)\right)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\gamma 1}}\right)\right)\right.$
$\displaystyle\hphantom{R_{j_{\alpha}k_{\beta}}=}{}+\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\beta 1}}\right)\right)\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\gamma 1}}\right)\right)$
$\displaystyle\left.\hphantom{R_{j_{\alpha}k_{\beta}}=}{}-\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\gamma 1}}\right)\right)\partial_{j_{\alpha}}\left(\log\left(\frac{\det
S}{s^{\gamma 1}}\right)\right)\right].$ (6.24)

We observe that since the cofactors $s^{\alpha 1}$ and $s^{\beta 1}$ are
independent of the groups of variables ${\bf x}^{\alpha}$ and ${\bf
x}^{\beta}$ respectively and since $\sum\limits_{\alpha=1}^{r}l_{\alpha}=n$,
we have

$\displaystyle-\frac{1}{2}\sum_{\gamma=1}^{r}l_{\gamma}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\log\left(\frac{\det
S}{s^{\gamma
1}}\right)\right)+\partial_{j_{\alpha}}\partial_{k_{\beta}}\big{(}\log(\det
S)\big{)}$
$\displaystyle\qquad{}=-\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right].$

We write

$\displaystyle-\frac{1}{2}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right]$
$\displaystyle\qquad{}=-\frac{3}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right]+\frac{1}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right],$
(6.25)

and compute the second term in (6.25) as follows

$\displaystyle\frac{1}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right]$
$\displaystyle\qquad{}=\frac{1}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left[\sum_{1\leq\gamma\neq\alpha,\beta\leq
r}\log\left(\frac{(\det S)^{l_{\gamma}}}{\big{(}s^{\gamma
1}\big{)}^{l_{\gamma}}}\right)+\log\left(\frac{(\det
S)^{l_{\alpha}+l_{\beta}-2}}{s^{\alpha 1}s^{\beta 1}}\right)\right].$

Evaluating the derivatives an using the fact that the cofactors $s^{\alpha 1}$
and $s^{\beta 1}$ are independent of the groups of variables ${\bf
x}^{\alpha}$ and ${\bf x}^{\alpha}$, we conclude that the expression (6.24) of
the off-block diagonal Ricci curvature can be written as

$\displaystyle
R_{j_{\alpha}k_{\beta}}=-\frac{3}{4}\partial_{j_{\alpha}}\partial_{k_{\beta}}\log\left[\frac{(\det
S)^{n-2}}{\big{(}s^{11}\big{)}^{l_{1}}\cdots\big{(}s^{r1}\big{)}^{l_{r}}}\right]+\frac{1}{4}(l_{\alpha}+l_{\beta}-2)\frac{s^{\alpha
1}s^{\beta 1}}{\det
S}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{\det S}{s^{\alpha
1}s^{\beta 1}}\right)$
$\displaystyle\hphantom{R_{j_{\alpha}k_{\beta}}=}{}+\sum_{\gamma\neq\alpha,\beta=1}^{r}l_{\gamma}\left(\partial_{j_{\alpha}}\log\frac{s^{\gamma
1}}{\det S}\right)\left(\partial_{k_{\beta}}\log\frac{s^{\alpha 1}}{\det
S}\right)+\left(\partial_{j_{\alpha}}\log\frac{s^{\beta 1}}{\det
S}\right)\left(\partial_{k_{\beta}}\log\frac{s^{\gamma 1}}{\det S}\right)$
$\displaystyle\hphantom{R_{j_{\alpha}k_{\beta}}=}{}-\frac{\det S}{s^{\gamma
1}}\partial_{j_{\alpha}}\partial_{k_{\beta}}\left(\frac{s^{\gamma 1}}{\det
S}\right).$

This completes the proof of Theorem 1.4.

###  6.2 Proof of Theorem 1.5

Our proof follows the structure of the one given in [3] for the special case
of Theorem 1.5 corresponding to Stäckel metrics satisfying the classical
Robertson conditions (1.7). We shall begin from the general expression for the
commutator of two operators of the form

$\Delta_{K_{(\alpha)}}=\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}+\sum_{\gamma=1}^{r}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}B_{(\alpha)}^{j_{\gamma}}\partial_{j_{\gamma}},$

and analyze this expression for the case where $\Delta_{K_{(\alpha)}}$ is
given by

$\Delta_{K_{(\alpha)}}=\nabla_{i}(K_{(\alpha)}^{ij}\nabla_{j})=\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\nabla_{i_{\gamma}}\big{(}K_{(\alpha)}^{i_{\gamma}j_{\gamma}}\nabla_{j_{\gamma}}\big{)},$

where the $\big{(}K_{(\alpha)}^{ij}\big{)}$ are the Killing tensors defined by
(1.16). We shall then prove that the commutator is identically zero for all
Painlevé metrics satisfying the generalized Robertson conditions. We shall see
that the generalized Killing–Eisenhart equations (3.2) established in Lemma
3.1 play a key role in the analysis of the commutator.

We have

$\displaystyle\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=\sum_{\gamma,\epsilon=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{k_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\sum_{p_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\Big{\\{}2\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}\partial_{j_{\gamma}}\partial_{k_{\epsilon}}\partial_{p_{\epsilon}}$
$\displaystyle\hphantom{\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=}{}+\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}+B_{(\alpha)}^{j_{\beta}}\partial_{j_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}$
$\displaystyle\hphantom{\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=}{}-B_{(\beta)}^{j_{\beta}}\partial_{j_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}\partial_{k_{\epsilon}}\partial_{p_{\epsilon}}+2\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}B_{\beta}^{p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}B_{\alpha}^{p_{\epsilon}}\big{)}\partial_{j_{\gamma}}\partial_{p_{\epsilon}}$
$\displaystyle\hphantom{\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=}{}+\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}B_{(\beta)}^{p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}B_{(\alpha)}^{p_{\epsilon}}+B_{(\alpha)}^{j_{\gamma}}\partial_{j_{\gamma}}B_{(\beta)}^{p_{\epsilon}}$
$\displaystyle\hphantom{\big{[}\Delta_{K_{(\alpha)}},\Delta_{K_{(\beta)}}\big{]}=}{}-B_{(\beta)}^{j_{\gamma}}\partial_{j_{\gamma}}B_{(\alpha)}^{p_{\epsilon}}\big{)}\partial_{p_{\epsilon}}\Big{\\}}.$
(6.26)

We will now evaluate the coefficients of the third, second and first
derivatives in the expression (6.26) of the commutator for Painlevé metrics
satisfying the generalized Robertson conditions, and show that must vanish
identically.

Over the course of the calculations, it will be useful to rewrite the
expressions (1.16) of the block components of the Killing tensors
$\big{(}K_{(\alpha)}^{ij}\big{)}$ in the form

$\displaystyle
K_{(\alpha)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\gamma}\frac{s^{\gamma
1}}{\det
S}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\gamma}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}},$
(6.27)

where the quantities $\rho_{\alpha\gamma}$ are defined by

$\displaystyle\rho_{\alpha\gamma}=\frac{s^{\gamma\alpha}}{s^{\gamma 1}},$
(6.28)

in which case we obtain

$\displaystyle
A_{(\alpha)}^{i_{\gamma}j_{\gamma}}=K_{(\alpha)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\gamma}\frac{s^{\gamma
1}}{\det
S}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\gamma}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}},$
$\displaystyle
B_{(\alpha)}^{j_{\gamma}}=-\rho_{\alpha\gamma}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\Gamma_{i_{\gamma}},$
(6.29)

where

$\Gamma_{k_{\epsilon}}=-\frac{1}{2}\partial_{k_{\epsilon}}\log(|g|)-\sum_{p_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\sum_{h_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}(g_{\epsilon})_{k_{\epsilon}p_{\epsilon}}\partial_{h_{\epsilon}}(g^{\epsilon})^{h_{\epsilon}p_{\epsilon}}.$

Note that a standard calculation shows the important result:

######  Lemma 6.1.

The generalized Robertson conditions (1.20) are equivalent to

$\displaystyle\partial_{j_{\alpha}}\Gamma_{k_{\beta}}=0,\qquad\forall\,1\leq\alpha\neq\beta\leq
r.$

We will compute the expressions of the derivatives of the coefficients
$A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$ and $B_{(\alpha)}^{j_{\gamma}}$ when
needed during the calculations, making use of the generalized
Killing–Eisenhart equations (3.2).

We begin with the coefficients of the third derivatives in (6.26), whose
vanishing is equivalent to the condition

$\displaystyle\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}=0.$
(6.30)

We shall see shortly that in analogy with the Stäckel case, the vanishing of
these coefficients does not require the generalized Robertson conditions and
holds true for all Painlevé metrics. When the expression (1.16) of the Killing
tensors $\big{(}K_{(\alpha)}^{ij}\big{)}$ is substituted into the condition
(6.30), the latter reduces to

$\displaystyle\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\rho_{\alpha\gamma}\frac{s^{\gamma
1}}{\det
S}G^{i_{\gamma}j_{\gamma}}\left(\partial_{i_{\gamma}}\left(\rho_{\beta\epsilon}\frac{s^{\epsilon
1}}{\det
S}\right)\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}+\rho_{\beta\epsilon}\frac{s^{\epsilon
1}}{\det
S}\partial_{i_{\gamma}}\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}\right)$
$\displaystyle\qquad{}=\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\rho_{\beta\gamma}\frac{s^{\gamma
1}}{\det
S}G^{i_{\gamma}j_{\gamma}}\left(\partial_{i_{\gamma}}\left(\rho_{\alpha\epsilon}\frac{s^{\epsilon
1}}{\det
S}\right)\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}+\rho_{\alpha\epsilon}\frac{s^{\epsilon
1}}{\det
S}\partial_{i_{\gamma}}\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}\right).$
(6.31)

We now distinguish between the cases $\gamma=\epsilon$ and
$\gamma\neq\epsilon$ when analyzing (6.31). If $\gamma=\epsilon$, then the
derivatives of $G^{\epsilon}$ cancel out in (6.31) and using (6.28), the
condition (6.31) reduces to

$\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{(}G^{\gamma}\big{)}^{k_{\gamma}p_{\gamma}}\left(\frac{s^{\gamma\alpha}}{\det
S}\partial_{i_{\gamma}}\left(\frac{s^{\gamma\beta}}{\det
S}\right)-\frac{s^{\gamma\beta}}{\det
S}\partial_{i_{\gamma}}\left(\frac{s^{\gamma\alpha}}{\det S}\right)\right)=0.$

But the latter is an identity on account of the generalized Killing–Eisenhart
equations expressed in the form (3.4). Likewise, if $\gamma\neq\epsilon$, then
using the fact that we have then
$\partial_{i_{\gamma}}\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}=0$,
the condition (6.31) becomes

$\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{(}G^{\epsilon}\big{)}^{k_{\epsilon}p_{\epsilon}}\left(\frac{s^{\gamma\alpha}}{\det
S}\partial_{i_{\gamma}}\left(\frac{s^{\epsilon\beta}}{\det
S}\right)-\frac{s^{\gamma\beta}}{\det
S}\partial_{i_{\gamma}}\left(\frac{s^{\epsilon\alpha}}{\det
S}\right)\right)=0,$

which is again an identity by virtue of the generalized Killing–Eisenhart
equations expressed in the form (3.4).

We now proceed with the coefficients of the second derivatives in (6.26),
whose vanishing is equivalent to the conditions

$\displaystyle\sum_{i_{\gamma}=!_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}\big{)}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}+B_{(\alpha)}^{j_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}\big{)}-B_{(\beta)}^{j_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}\big{)}$
$\displaystyle\qquad{}+2\sum_{i_{\epsilon}=!_{\epsilon}}^{l_{\epsilon}}\big{(}A_{(\alpha)}^{i_{\epsilon}(k_{\epsilon}}\partial_{i_{\epsilon}}\big{(}B_{(\beta)}^{p_{\epsilon})}\big{)}-A_{(\beta)}^{i_{\epsilon}(k_{\epsilon}}\partial_{i_{\epsilon}}\big{(}B_{(\alpha)}^{p_{\epsilon})}\big{)}\big{)}=0,$
(6.32)

and

$\displaystyle\sum_{i_{\gamma}=!_{\gamma}}^{l_{\gamma}}\big{(}A_{(\alpha)}^{i_{\gamma}(j_{\gamma}}\partial_{i_{\gamma}}\big{(}B_{(\beta)}^{p_{\epsilon})}\big{)}-A_{(\beta)}^{i_{\gamma}(j_{\gamma}}\partial_{i_{\gamma}}\big{(}B_{(\alpha)}^{p_{\epsilon})}\big{)}\big{)}=0,$
(6.33)

where the round brackets denote symmetrization of indices.

We begin with the condition (6.32), for which we will need expressions for the
first and second derivatives of $A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$ and
$B_{(\alpha)}^{j_{\gamma}}$. Using (6.27) and (6.29), we obtain

$\partial_{k_{\epsilon}}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}=\left(\partial_{k_{\epsilon}}(\rho_{\alpha\gamma})\frac{s^{\gamma
1}}{\det S}+\rho_{\alpha\gamma}\partial_{k_{\epsilon}}\left(\frac{s^{\gamma
1}}{\det
S}\right)\right)\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}+\rho_{\alpha\gamma}\frac{s^{\gamma
1}}{\det
S}\partial_{k_{\epsilon}}\big{(}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{)}{\delta^{\gamma}}_{\epsilon}.$

Using the generalized Killing–Eisenhart equations (3.2), the preceding
equation reduces to

$\displaystyle\partial_{k_{\epsilon}}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\epsilon}\partial_{k_{\epsilon}}\left(\frac{s^{\gamma
1}}{\det
S}\right)\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\\!+\rho_{\alpha\gamma}\frac{s^{\gamma
1}}{\det
S}\partial_{k_{\epsilon}}\big{(}\big{(}G^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{)}{\delta^{\gamma}}_{\epsilon}=\rho_{\alpha\epsilon}\partial_{k_{\epsilon}}\big{(}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{)}.\\!\\!\\!\\!$
(6.34)

Likewise, we obtain the following expressions for the second derivatives of
the coefficients $A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$ by using again the
generalized Killing–Eisenhart equations (3.2)

$\displaystyle\partial_{k_{\epsilon}}\partial_{p_{\epsilon}}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}=\rho_{\alpha\epsilon}\partial_{k_{\epsilon}}\partial_{p_{\epsilon}}\big{(}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\big{)}.$
(6.35)

This implies immediately that the first two terms in (6.32) cancel each other
out, that is

$\displaystyle\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}\big{)}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{)}\big{)}$
$\displaystyle\qquad{}=\sum_{i_{\gamma}=!_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{[}\rho_{\alpha\gamma}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\rho_{\beta\gamma}\partial_{i_{\gamma}}\partial_{j_{\gamma}}(g^{\epsilon})^{k_{\epsilon}p_{\epsilon}}-\rho_{\beta\gamma}\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\rho_{\alpha\gamma}\partial_{i_{\gamma}}\partial_{j_{\gamma}}(g^{\epsilon})^{k_{\epsilon}p_{\epsilon}}\big{]}=0.\\!\\!\\!\\!$
(6.36)

By making use of the preceding remark, by observing that

$\displaystyle
B_{(\alpha)}^{j_{\gamma}}=-\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\Gamma_{i_{\gamma}},$
(6.37)

and by using the expressions (6.34) and (6.35) for the first and second
derivatives of the coefficients $A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$, the
condition (3.7) becomes

$\displaystyle\sum_{i_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\sum_{h_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\big{(}A_{(\alpha)}^{i_{\epsilon}k_{\epsilon}}A_{(\beta)}^{h_{\epsilon}p_{\epsilon}}+A_{(\alpha)}^{i_{\epsilon}p_{\epsilon}}A_{(\beta)}^{h_{\epsilon}k_{\epsilon}}-A_{(\alpha)}^{h_{\epsilon}p_{\epsilon}}A_{(\beta)}^{i_{\epsilon}k_{\epsilon}}-A_{(\alpha)}^{h_{\epsilon}k_{\epsilon}}A_{(\beta)}^{i_{\epsilon}p_{\epsilon}}\big{)}\partial_{i_{\epsilon}}\Gamma_{h_{\epsilon}}$
$\displaystyle{}+\\!\sum_{i_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\sum_{h_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\\!\big{(}A_{(\alpha)}^{i_{\epsilon}k_{\epsilon}}\partial_{i_{\epsilon}}(A_{(\beta)}^{h_{\epsilon}p_{\epsilon}})+A_{(\alpha)}^{i_{\epsilon}p_{\epsilon}}\partial_{i_{\epsilon}}(A_{(\beta)}^{h_{\epsilon}k_{\epsilon}})-A_{(\beta)}^{i_{\epsilon}k_{\epsilon}}\partial_{i_{\epsilon}}(A_{(\alpha)}^{h_{\epsilon}p_{\epsilon}})-A_{(\beta)}^{i_{\epsilon}p_{\epsilon}}\partial_{i_{\epsilon}}(A_{(\alpha)}^{h_{\epsilon}k_{\epsilon}})\big{)}\Gamma_{h_{\epsilon}}=0.$

It is now easily verified that the second double sum in the preceding
expression vanishes identically as a consequence of (6.36), and that the first
double sum is identically zero upon substitution of the expressions (6.29) of
the coefficients $A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$.

Next we turn our attention to the condition (6.33), which we may rewrite,
using (6.37), as

$\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{h_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}A_{(\beta)}^{h_{\epsilon}p_{\epsilon}}-A_{(\alpha)}^{h_{\epsilon}p_{\epsilon}}A_{(\beta)}^{i_{\gamma}j_{\gamma}}\big{)}\big{(}\partial_{i_{\gamma}}\Gamma_{h_{\epsilon}}-\partial_{h_{\epsilon}}\Gamma_{i_{\gamma}}\big{)}=0.$

We have

$\displaystyle\partial_{i_{\gamma}}\Gamma_{h_{\epsilon}}-\partial_{h_{\epsilon}}\Gamma_{i_{\gamma}}=-\partial_{i_{\gamma}}\partial_{h_{\epsilon}}\left(\log\frac{s^{\epsilon
1}}{\det
S}\right)+\partial_{h_{\epsilon}}\partial_{i_{\gamma}}\left(\log\frac{s^{\gamma
1}}{\det S}\right)=0,$ (6.38)

since the factors of $\det S$ cancel out in the logarithmic derivatives, and
since $s^{\epsilon 1}$ (resp. $s^{\gamma 1}$) is independent of the groups of
variables ${\bf x}^{\epsilon}$ (resp. ${\bf x}^{\gamma}$). Finally, we must
show that the coefficients of the first derivatives in (6.26) are identically
zero. We shall see that the analysis of these coefficients is slightly more
involved than that of the second and third derivatives, and that the argument
needed to prove their vanishing makes use of the generalized Robertson
conditions.

Using (6.37) the vanishing of the coefficients of the first derivatives in
(6.26) is seen to be equivalent to

$\displaystyle\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{k_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\big{(}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}A_{(\beta)}^{i_{\gamma}j_{\gamma}}\big{)}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\Gamma_{k_{\epsilon}}$
$\displaystyle\qquad{}+\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{k_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\big{(}\big{[}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}\partial_{j_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{]}\Gamma_{k_{\epsilon}}$
$\displaystyle\qquad{}+2\big{[}A_{(\alpha)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}-A_{(\beta)}^{i_{\gamma}j_{\gamma}}\partial_{i_{\gamma}}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\big{]}\big{)}\partial_{j_{\gamma}}\Gamma_{k_{\epsilon}}$
$\displaystyle\qquad{}-\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{k_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}\sum_{m_{\gamma}=1_{\gamma}}^{l_{\gamma}}\big{(}A_{(\alpha)}^{m_{\gamma}j_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\beta)}^{k_{\epsilon}p_{\epsilon}}\Gamma_{k_{\epsilon}}\big{)}-A_{(\beta)}^{m_{\gamma}j_{\gamma}}\partial_{j_{\gamma}}\big{(}A_{(\alpha)}^{k_{\epsilon}p_{\epsilon}}\Gamma_{k_{\epsilon}}\big{)}\big{)}=0.$
(6.39)

The second triple sum in (6.39) vanishes identically on account of (6.30) and
(6.36). Likewise, using again (6.30) and the expressions (6.29) for
$A_{(\alpha)}^{i_{\gamma}j_{\gamma}}$, the condition (6.39) reduces to

$\displaystyle\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}\sum_{k_{\epsilon}=1_{\epsilon}}^{l_{\epsilon}}(\rho_{\alpha\gamma}\rho_{\beta\epsilon}-\rho_{\alpha\epsilon}\rho_{\beta\gamma})\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}(g^{\epsilon})^{k_{\epsilon}p_{\epsilon}}\big{(}\partial_{i_{\gamma}}\partial_{j_{\gamma}}\Gamma_{k_{\epsilon}}-\Gamma_{i_{\gamma}}(\partial_{j_{\gamma}}\Gamma_{k_{\epsilon}})\big{)}=0.$
(6.40)

In analogy with the notation used in [3], we introduce the tensor
$\big{(}{C^{i}}_{j}\big{)}$ defined by

$\displaystyle{C^{i_{\gamma}}}_{k_{\epsilon}}=\sum_{j_{\gamma}=1_{\gamma}}^{l_{\gamma}}(\rho_{\alpha\gamma}\rho_{\beta\epsilon}-\rho_{\alpha\epsilon}\rho_{\beta\gamma})\big{(}g^{\gamma}\big{)}^{i_{\gamma}j_{\gamma}}\partial_{j_{\gamma}}\Gamma_{k_{\epsilon}}.$
(6.41)

The final steps in the proof will be to show that (6.40) is satisfied if an
only if the tensor $\big{(}{C^{i}}_{j}\big{)}$ has zero divergence, that is

$\displaystyle\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\nabla_{i_{\gamma}}{C^{i_{\gamma}}}_{k_{\epsilon}}=0,$

and that the generalized Robertson conditions are equivalent to
${C^{i_{\gamma}}}_{k_{\epsilon}}=0$. These are the analogues for Painlevé
metrics of the steps followed in [3] for the proof of the corresponding result
for the special case of Stäckel metrics satisfying the classical Robertson
conditions.

We have

$\displaystyle\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\nabla_{i_{\gamma}}{C^{i_{\gamma}}}_{k_{\epsilon}}=\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\left(\partial_{i_{\gamma}}{C^{i_{\gamma}}}_{k_{\epsilon}}+\sum_{\delta=1}^{r}\sum_{h_{\delta}=1_{\delta}}^{l_{\delta}}{\Gamma^{i_{\gamma}}}_{j_{\gamma}h_{\delta}}{C^{h_{\delta}}}_{k_{\epsilon}}\right.$
$\displaystyle\left.\hphantom{\sum_{\gamma=1}^{r}\sum_{i_{\gamma}=1_{\gamma}}^{l_{\gamma}}\nabla_{i_{\gamma}}{C^{i_{\gamma}}}_{k_{\epsilon}}=}{}-\sum_{\delta=1}^{r}\sum_{p_{\delta}=1_{\delta}}^{l_{\delta}}{\Gamma^{p_{\delta}}}_{j_{\gamma}k_{\epsilon}}{C^{h_{\delta}}}_{k_{\epsilon}}\right).$
(6.42)

Substituting into the expression (6.42) of the divergence the definition
(6.41) of the tensor $\big{(}{C^{i}}_{j}\big{)}$ and the expressions (6.4) and
(6.5) that were computed above for the Christoffel symbols of Painlevé
metrics, we obtain