Bi-Hamiltonian representation of Stäckel systems

It is shown that a linear separation relations are fundamental objects for integration by quadratures of Stäckel separable Liouville integrable systems (the so-called Stäckel systems). These relations are further employed for the classification of Stäckel systems. Moreover, we prove that {\em any} Stäckel separable Liouville integrable system can be lifted to a bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other known result this implies that the existence of bi-Hamiltonian representation of Liouville integrable systems is a necessary condition for Stäckel separability.

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