  We address the problem of the separation of variables for the Hamilton-Jacobi
equation within the theoretical scheme of bi-Hamiltonian geometry. We use the
properties of a special class of bi-Hamiltonian manifolds, called omega-N
manifolds, to give intrisic tests of separability (and Staeckel separability)
for Hamiltonian systems. The separation variables are naturally associated with
the geometrical structures of the omega-N manifold itself. We apply these
results to bi-Hamiltonian systems of the Gel'fand-Zakharevich type and we give
explicit procedures to find the separated coordinates and the separation
relations.
