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A family of classical integrable systems defined on a deformation of thetwo-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructedthrough Hamiltonians defined on the non-standard quantum deformation of a sl(2)Poisson coalgebra. All these spaces have a non-constant curvature that dependson the deformation parameter z. As particular cases, the analogues of theharmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed.Another deformed Hamiltonian is also shown to provide superintegrable systemson the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constantcurvature that exactly coincides with z. According to each specific space, theresulting potential is interpreted as the superposition of a central harmonicoscillator with either two more oscillators or centrifugal barriers. Thenon-deformed limit z=0 of all these Hamiltonians can then be regarded as thezero-curvature limit (contraction) which leads to the corresponding(super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde

Integrable potentials on spaces with curvature from quantum groups