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Painlev\'e metrics are a class of Riemannian metrics which generalize thewell-known separable metrics of St\"ackel to the case in which the additiveseparation of variables for the Hamilton-Jacobi equation is achieved in termsof groups of independent variables rather than the complete orthogonalseparation into ordinary differential equations which characterizes theSt\"ackel case. Our goal in this paper is to carry out for Painlev\'e metricsthe generalization of the analysis, which has been extensively performed in theSt\"ackel case, of the relation between separation of variables for theHamilton-Jacobi and Helmholtz equations, and of the connections betweenquadratic first integrals of the geodesic flow and symmetry operators for theLaplace-Beltrami operator. We thus obtain the generalization for Painlev\'emetrics of the Robertson separability conditions for the Helmholtz equationwhich are familiar from the St\"ackel case, and a formulation thereof in termsof the vanishing of the off-block diagonal components of the Ricci tensor,which generalizes the one obtained by Eisenhart for St\"ackel metrics. We alsoshow that when the generalized Robertson conditions are satisfied, there exist$r<n$ linearly independent second-order differential operators which commutewith the Laplace-Beltrami operator and which are mutually commuting. Theseoperators admit the block-separable solutions of the Helmholtz equation asformal eigenfunctions, with the separation constants as eigenvalues. Finally,we study conformal deformations which are compatible with the separation intoblocks of variables of the Helmholtz equation for Painlev\'e metrics, leadingto solutions which are $R$-separable in blocks. The paper concludes with a setof open questions and perspectives.

Separability and Symmetry Operators for Painlev\'e Metrics and their Conformal Deformations