Separation of variables for bi-Hamiltonian systems

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 omegaN manifolds, to give intrisic tests of separability ( and Stackel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the omegaN 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