Geodesic equivalence via integrability

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor g(ialpha)(g) over bar (alphaj) are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Lionville integrable. This gives us topological obstacles to geodesic equivalence.