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

For a magnetic geodesic flow on the 2-torus the only known integrable example is that of a flow integrable for all energy levels. It has an integral linear in momenta and corresponds to a one parameter group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on a single energy level is considered. Then, in addition to the example mentioned above, a few other explicit examples with quadratic in momenta integrals can be constructed by means of the Maupertuis’ principle. Recently we proved that such an integrability problem can be reduced to a remarkable semi-Hamiltonian system of quasi-linear PDEs and to the question of the existence of smooth periodic solutions for this system. Our main result of the present paper states that any Liouville metric with the zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with a small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of an integral quadratic in momenta.