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 admitsan additional high degree first integral polynomial in momenta and is independentof the energy integral. In an earlier work by the first two authors, it wasannounced that if such integral is preserved at a sufficiently many different energy levelsthen there necessarily exists a linear integral at allenergy levels. The proof of the announce was incomplete.Here we finish the proof of the above assertion.