  The St\"ackel transform is applied to the geodesic motion on Euclidean space,
through the harmonic oscillator and Kepler-Coloumb potentials, in order to
obtain maximally superintegrable classical systems on N-dimensional Riemannian
spaces of nonconstant curvature. By one hand, the harmonic oscillator potential
leads to two families of superintegrable systems which are interpreted as an
intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the
so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential
gives rise to an oscillator system on a spherical curved space as well as to
the Taub-NUT oscillator. Their integrals of motion are explicitly given. The
role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector for
all of these Hamiltonians is highlighted throughout the paper. The
corresponding quantum maximally superintegrable systems are also presented.
