The perturbed Kepler problem is shown to be a bi-Hamiltonian system in spite of the fact that the graph of the Hamilton function is not a hypersurface of translation, which goes against a necessary condition for the existence of the bi-Hamiltonian structure according to the Fernandes theorem. In fact, the initial and perturbed Kepler systems are both isochronous systems and, therefore, the Fernandes theorem cannot be applied to them.