SIGMA 7 (2011), 048, 15 pages arXiv:1103.4554 http://dx.doi.org/10.3842/SIGMA.2011.048 Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)” Ángel Ballesteros a, Alberto Enciso b, Francisco J. Herranz a, Orlando Ragnisco c and Danilo Riglioni c a) Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain b) Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UC3M), Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 14-16, E-28049 Madrid, Spain c) Dipartimento di Fisica, Università di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy Abstract 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. Key words: coupling constant metamorphosis, integrable systems, curvature, harmonic oscillator, Kepler-Coulomb, Fradkin tensor, Laplace-Runge-Lenz vector, Taub-NUT, Darboux surfaces. Source.