The Post-Winternitz system on spherical and hyperbolic spaces: A proof of the superintegrability making use of complex functions and a curvature-dependent formalism

Two important advances in integrability have been the recent discovery of the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system (related to the harmonic oscillator) and the Post-Winternitz system (related to the Kepler problem). The properties of the TTW system have been recently studied on the two-dimensional spherical Sκ2 (κ>0) and hyperbolic Hκ2 (κ<0) spaces by making use of a curvature-dependent formalism and the existence of a complex factorization for the higher-order constant of motion. Now in this Letter we prove that a similar technique can also be applied for the study of the PW system.