# American Institute of Mathematical Sciences

November  2019, 39(11): 6565-6583. doi: 10.3934/dcds.2019285

## Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels

 1 Sobolev Institute of Mathematics, Novosibirsk, 4 Acad. Koptyug avenue, 630090, Russia 2 Novosibirsk State University, Novosibirsk, 1 Pirogova str., 630090, Russia

* Corresponding author: Sergei Agapov

Received  January 2019 Published  August 2019

Fund Project: The first author is supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation)

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral $F$ on $N+2$ different energy levels which is polynomial in momenta of an arbitrary degree $N$ with analytic periodic coefficients. It is proved that in this case the magnetic field and the metric are functions of one variable and there exists a linear in momenta first integral on all energy levels.

Citation: Sergei Agapov, Alexandr Valyuzhenich. Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6565-6583. doi: 10.3934/dcds.2019285
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