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The St\"ackel transform is applied to the geodesic motion on Euclidean space,through the harmonic oscillator and Kepler-Coloumb potentials, in order toobtain maximally superintegrable classical systems on N-dimensional Riemannianspaces of nonconstant curvature. By one hand, the harmonic oscillator potentialleads to two families of superintegrable systems which are interpreted as anintrinsic Kepler-Coloumb system on a hyperbolic curved space and as theso-called Darboux III oscillator. On the other, the Kepler-Coloumb potentialgives rise to an oscillator system on a spherical curved space as well as tothe Taub-NUT oscillator. Their integrals of motion are explicitly given. Therole of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector forall of these Hamiltonians is highlighted throughout the paper. Thecorresponding quantum maximally superintegrable systems are also presented.

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the St\"ackel Transform