The classical Smorodinsky-Winternitz systems on the /VD sphere, Euclidean and hyperbolic spaces SN, E^ and H^ are simultaneously approached starting from the Lie algebras so^fA' + l), which include a parametric dependence on the curvature. General expressions for the Hamiltonian and its integrals of motion are given in terms of intrinsic geodesic coordinate systems. Bach Lie algebra generator gives rise to an integral of motion, so that a set of N(N + l)/2 integrals is obtained. Furthermore, 2N - 1 functionally independent ones are identified which, in turn, shows that the well-known maximal superintegrability of the Smorodinsky-Winternitz system on E^ is preserved when curvature arises. On both SN and H^, the resulting system can be interpreted as a superposition of an "actual" oscillator and N "ideal" oscillators (for the sphere, these are alike the actual ones), which can also be understood as Ar "centrifugal terms"; this is the form seen in the Euclidean limiting case.
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