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 Hamiltonian systems of low dimensions using the Hamilton–Jacobi method of separation of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems defined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found.