Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space

We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.