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In this paper we deal with the classical question of existence of polynomialin momenta integrals for geodesic flows on the 2-torus. For the quasi-linearsystem on coefficients of the polynomial integral we consider the region (socalled elliptic regions) where there are complex-conjugate eigenvalues. We showthat for quartic integrals in the other two eigenvalues are real and genuinelynonlinear. This observation together with the property of the system to be Rich(Semi-Hamiltonian) enables us to classify elliptic regions completely. The caseof complex-conjugate eigenvalues for the system corresponding to the integralof degree 3 is done similarly. These results show that if new integrableexamples exist they could be found only within the region of Hyperbolicity ofthe quasi-linear system.

Cubic and Quartic integrals for geodesic flow on 2-torus via system of Hydrodynamic type