Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus (Q1282506)

Systems whose configuration space is the two-dimensional torus and the Hamiltonian contains quadratic and linear terms in the momenta are considered. Conditions for the existence of a second independent integral, linear or quadratic in the momenta, are found. It is also shown that, assuming that the metric tensor is a trigonometric polynomial with respect to the angle coordinates, if the geodesic flow on the torus admits a polynomial integral independent of the Hamiltonian, then there exists a polynomial integral of degree 1 or 2.