Abstract
An infinite family of quasi-maximally superintegrable Hamiltonians with a com- mon set of (2N 3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. Moreover, another Hamil- tonian in this family is shown to generate geodesic motions on Riemannian and relativistic spaces all of whose sectional curvatures are constant and equal to the deformation parame- ter z. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry.
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