The St��ckel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. In the present paper we consider a class of St��ckel separable systems where the entries of the St��ckel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the St��ckel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems (that is, they admit separation of variables in the Schr��dinger equation) in the class under study.

13 pages, to appear in Phys. Lett. A