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In this paper the geodesic flow on a 2-torus in a non-zero magnetic field isconsidered. Suppose that this flow admits an additional first integral $F$ on$N+2$ different energy levels which is polynomial in momenta of arbitrarydegree $N$ with analytic periodic coefficients. It is proved that in this casethe magnetic field and metrics are functions of one variable and there exists alinear in momenta first integral on all energy levels.

Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels