Polynomial separable indefinite natural systems

We review the conditions for separability of 2-dimensional indefinite natural Hamiltonian systems. We examine the possibility that the separability condition is satisfied on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have separability at arbitrary values of the Hamiltonian (strong integrability). We give a list of separable polynomial potentials and discuss the kind of separable structures they admit.