  We show that with every separable calssical Stackel 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 system 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 conditions under which a given flat bi-cofactor
system can be deformed to a family of geodesically equivalent flat bi-cofactor
systems.
