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

On bi-Hamiltonian formulation of the perturbed Kepler problem