Superintegrable dynamics on $H^2$ generated by coupling the Morse and Rosen-Morse potentials

A Hamiltonian dynamics defined on the two-dimensional hyperbolic plane by coupling the Morse and Rosen-Morse potentials is analyzed. It is demonstrated that orbits of all bounded motions are closed iff the product of the parameter $\tilde{a}$ of the Morse potential and the square root of the absolute value of the curvature is a rational number. This property of trajectories equivalent to the maximal superintegrability is confirmed by explicit construction of polynomial superconstant of motion.