Integrable geodesic flows on 2-torus: Formal solutions and variational principle

In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in Bialy and Mironov (2011) that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metric of Egorov type. We use this fact in order to prove that in the case of integrals of degree three and four the system is in fact equivalent to a single remarkable equation of order 3 and 4 respectively. Remarkably the equation for the case of degree four has variational meaning: it is Euler–Lagrange equation of a variational principle. Next we prove that this equation for n=4n=4 has formal double periodic solutions as a series in a small parameter.