New Semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

TL;DR

This paper investigates magnetic geodesic flows on the 2-torus, showing that the existence of polynomial first integrals leads to a Semi-Hamiltonian system of equations, revealing new integrability structures.

Contribution

It establishes a connection between polynomial first integrals of magnetic flows and Semi-Hamiltonian systems, introducing a novel hierarchy related to integrable magnetic flows on surfaces.

Findings

Existence of polynomial first integrals implies a Semi-Hamiltonian system.

The system admits Riemann invariants in hyperbolic regions.

The equations can be expressed in conservation laws form.

Abstract

We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.