  The only one example has been known of magnetic geodesic flow on the 2-torus
which has a polynomial in momenta integral independent of the Hamiltonian. In
this example the integral is linear in momenta and corresponds to a one
parametric group preserving the Lagrangian function of the magnetic flow. In
this paper the problem of integrability on one energy level is considered. This
problem can be reduced to a remarkable Semi-hamiltonian system of quasi-linear
PDEs and to the question of existence of smooth periodic solutions for this
system. Our main result states that the pair of Liouville metric with zero
magnetic field on the 2-torus can be analytically deformed to a Riemannian
metric with small magnetic field so that the magnetic geodesic flow on an
energy level is integrable by means of a quadratic in momenta integral. Thus
our construction gives a new example of smooth periodic solution to the
Semi-hamiltonian (Rich) quasi-linear system of PDEs.
