First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus (Q1118220)

Let \(H=(p^ 2_ 1+p^ 2_ 2)/2+V(x,y)\) be a Hamiltonian function on a torus with the Cartesian coordinates \(x,y\). The author finds the functions \(V\) such that first integrals which are polynomials of degree \(1,2,3,4\) in \((p_ 1,p_ 2)\) exist. These functions are the same for degrees \(3,1\) on the one hand, and \(4,2\) on the other.