When is a Hamiltonian system separable?

The Hamiltonian system given by H= 1/2 p2+V(q) with V∈C∞(Rn) is considered. A method for integrating such a system is that of separating the variables in the Hamilton–Jacobi equation. It is known that if such a separation is possible, then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these. A criterion is proposed for deciding if separation is possible, and if it is, in which degeneration of elliptic coordinates it takes place.