A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

We propose a geometrical approach to the problem of integrability of Hamilt onian systems of low dimensions using the Hamilton-Jacobi method of separat ion of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems d efined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found. (C) 2001 Elsevier Science B.V All rights reserved.