  The geometric approach to mechanics based on the Jacobi metric allows to
easily construct natural mechanical systems which are integrable (actually
separable) at a fixed value of the energy. The aim of the present paper is to
investigate the dynamics of a simple prototype system outside the zero-energy
hypersurface. We find that the general situation is that in which integrability
is not preserved at arbitrary values of the energy. The structure of the
Hamiltonian in the separating coordinates at zero energy allows a perturbation
treatment of this system at energies slightly different from zero, by which we
obtain an analytical proof of non-integrability.
