  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 two-sphere, 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.
