Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

The Stäckel 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.