## Abstract

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= 4 has formal double periodic solutions as a series in a small parameter.

Original language | English |
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Pages (from-to) | 39-47 |

Number of pages | 9 |

Journal | Journal of Geometry and Physics |

Volume | 87 |

DOIs | |

State | Published - 1 Jan 2015 |

## Keywords

- Conservation laws
- Geodesic flows
- Integrable Hamiltonians
- Semi-Hamiltonian systems

## All Science Journal Classification (ASJC) codes

- Geometry and Topology
- General Physics and Astronomy
- Mathematical Physics