Complete sets of invariants for dynamical systems that admit a separation of variables

Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton-Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P₁ = H, P₂,..., P_n are the other second-order constants of the motion associated with the separable coordinates, and {Q_i, Q_j} = {P_i, P_j} = 0, {Q_i, P_j} = δ_ij. The 2n-1 functions Q₂,..., Q_n, P₁,..., P_n form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Q_j is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton-Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.