  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.
