  A unified algebraic construction of the classical Smorodinsky-Winternitz
systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie
groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions
for the Hamiltonian and its integrals of motion are given in a linear ambient
space $R^{N+1}$, and secondly they are expressed in terms of two geodesic
coordinate systems on the ND spaces themselves, with an explicit dependence on
the curvature as a parameter. On the sphere, the potential is interpreted as a
superposition of N+1 oscillators. Furthermore each Lie algebra generator
provides an integral of motion and a set of 2N-1 functionally independent ones
are explicitly given. In this way the maximal superintegrability of the ND
Euclidean Smorodinsky-Winternitz system is shown for any value of the
curvature.
