Superintegrability on the hyperbolic plane with integrals of any degree ≥2

Abstract

Starting from the work of Koenigs on superintegrable geodesic flows with one linear and two quadratic integrals, Matveev and Shevchishin have generalized it to the case of two cubic integrals. In this article superintegrability is achieved still keeping one linear integral but adding two extra integrals of any degree above two in the momenta. These extra integrals may exhibit either a trigonometric dependence in the Killing coordinate (a case we have already analyzed in [12] and is related to Zoll metrics which may be defined on the two-sphere) or an hyperbolic dependence. This companion case is studied here, revealing a geodesic flow which is never defined on the two-sphere, as opposed to the trigonometric case. Nevertheless it is possible to find some sufficient conditions under which the geodesic flow is defined on the hyperbolic plane.