Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

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.