  In this article we prove that many Hamiltonian systems that can not be
separably quantized in the classical approach of Robertson and Eisenhardt can
be separably quantized if we extend the class of admissible quantizations
through a suitable choice of Riemann space adapted to the Poisson geometry of
the system. Actually, in this article we prove that for every quadratic in
momenta St\"ackel system (defined on a 2n-dimensional Poisson manifold) for
which the St\"ackel matrix consists of monomials in position coordinates there
exist infinitely many quantizations - parametrized by n arbitrary functions -
that turn this system into a quantum separable St\"ackel system.
