  An infinite family of quasi-maximally superintegrable Hamiltonians with a
common 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 Hamiltonian 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 parameter z. This approach can be generalized to arbitrary
dimension by making use of coalgebra symmetry.
