Superintegrable geodesic flows on the hyperbolic plane
Abstract
In the framework laid down by Matveev and Shevchishin, superintegrability is achieved with one integral linear in the momenta (a Killing vector) and two extra integrals of of any degree above two in the momenta. However these extra integrals may exhibit either a trigonometric dependence in the Killing coordinate (a case we have already solved) or a hyperbolic dependence and this case is solved here. Unfortunately the resulting geodesic flow is {\em never} defined on the twosphere, as was the case for Koenigs systems (with quadratic extra integrals). Nevertheless we give some sufficient conditions under which the geodesic flow is defined on the hyperbolic plane.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.02703
 arXiv:
 arXiv:2110.02703
 Bibcode:
 2021arXiv211002703V
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 {\tt 32C05};
 {\tt 53C22};
 {\tt 37E99};
 {\tt 37J35};
 {\tt 37K25};
 {\tt 81V99}
 EPrint:
 32 pages no figure