New Semi-Hamiltonian hierarchy related to integrable magnetic flows on   surfaces
  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. 
Michael (Misha) Bialy and Andrey Mironov
math-ph math.MP nlin.SI