Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space

The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree $n$ independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for $n=3$ (which generalizes a theorem of Byalyi), and for $n=4$, $5$, and $6$ this is proved under some additional assumptions about the spectrum of the potential.