Classical and quantum superintegrable systems on the sphere and the hyperbolic 2-space

We present two superintegrable Hamiltonian systems in two dimensions, defined on the sphere and on the hyperbolic plane. These systems are generalised à la Tremblay-Turbiner-Winternitz (TTW), involving the introduction of a real parameter k > 0, with the aim of extending superintegrable Hamiltonian systems to curved spaces in a way similar to the TTW system on the plane. We carry out both classical and quantum analyses of these new systems. We prove that the superintegrability of the initial systems (i.e. when k = 1) is preserved when k is rational, as in the TTW case. A detailed study of their classical counterparts and trajectories is also included.