Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torus

We study the magnetic geodesic flow on the two-dimensional torus which admits an additional high degree first integral polynomial in momenta and is independent of the energy integral. In an earlier work by the first two authors, it was announced that if such integral is preserved at a sufficiently many different energy levels then there necessarily exists a linear integral at all energy levels. The proof of the announce was incomplete. Here we finish the proof of the above assertion.