  It is proved that the motion of a charge particle on a hyperbolic oriented
two-dimensional surface in a magnetic field given by the volume form of the
hyperbolic metric is completely integrable on the energy levels E < 1/2 in
terms of real-analytic integrals. However it was known that on the level E=1/2
every trajectory is transitive and the restrictions of this flow onto the
levels E>1/2 are Anosov flows
