Cubic and Quartic integrals for geodesic flow on 2-torus via system of   Hydrodynamic type
  In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for quartic integrals in the other two eigenvalues are real and genuinely nonlinear. This observation together with the property of the system to be Rich (Semi-Hamiltonian) enables us to classify elliptic regions completely. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist they could be found only within the region of Hyperbolicity of the quasi-linear system. 
Michael (Misha) Bialy and Andrey Mironov
math.DG