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In this paper we study quasi-linear system of partial differential equationswhich describes the existence of the polynomial in momenta first integral ofthe integrable geodesic flow on 2-torus. We proved in [3] that this is asemi-Hamiltonian system and we show here that the metric associated with thesystem is a metric of Egorov type. We use this fact in order to prove that inthe case of integrals of degree three and four the system is in fact equivalentto a single remarkable equation of order 3 and 4 respectively. Remarkably theequation for the case of degree four has variational meaning: it isEuler-Lagrange equation of a variational principle. Next we prove that thisequation for $n=4$ has formal double periodic solutions as a series in a smallparameter.

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