Energy dependent integrability

We review the conditions for separability of 2-dimensional 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 some new examples of systems admitting separating coordinates whose relation with the original ones explicitly depends on energy and provide a list of separable potentials discussing the nature of conserved quantities they admit.