Painlev�� metrics are a class of Riemannian metrics which generalize the well-known separable metrics of St��ckel 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 St��ckel case. Our goal in this paper is to carry out for Painlev�� metrics the generalization of the analysis, which has been extensively performed in the St��ckel 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 Painlev�� metrics of the Robertson separability conditions for the Helmholtz equation which are familiar from the St��ckel 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 St��ckel metrics. We also show that when the generalized Robertson conditions are satisfied, there exist $r