Non-Hamiltonian systems separable by Hamilton-Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.