Bi-Hamiltonian representation of St\"{a}ckel systems

It is shown that a linear separation relations are fundamental objects for integration by quadratures of St\"{a}ckel separable Liouville integrable systems (the so-called St\"{a}ckel systems). These relations are further employed for the classification of St\"{a}ckel systems. Moreover, we prove that {\em any} St\"{a}ckel 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 St\"{a}ckel separability.