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

## TL;DR

This paper proves that for magnetic geodesic flows on a 2-torus with polynomial first integrals on multiple energy levels, the magnetic field and metric depend on a single variable, ensuring a linear integral across all energies.

## Contribution

It establishes that the existence of polynomial first integrals on several energy levels implies the magnetic field and metric are one-variable functions, leading to a universal linear integral.

## Key findings

- Magnetic field and metric depend on one variable.
- Existence of polynomial first integrals on multiple energy levels.
- A linear first integral exists on all energy levels.

## Abstract

In this paper the geodesic flow on a 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 arbitrary degree $N$ with analytic periodic coefficients. It is proved that in this case the magnetic field and metrics are functions of one variable and there exists a linear in momenta first integral on all energy levels.