**Superintegrable potentials on N-dimensional Darboux spaces**

**Abstract:**

An infinite family of N-dimensional (ND) spaces endowed with an sl(2)-coalgebra symmetry is firstly introduced. Their geodesic flow is shown to
be superintegrable since all of them share a common set of (2N-3)
functionally independent integrals quadratic in the momenta which is obtained from the underlying sl(2)-coalgebra structure. In particular, ND spherically
symmetric spaces with Euclidean signature are shown to be sl(2)-coalgebra spaces. As a byproduct we obtain ND generalizations of the classical Darboux
surfaces, thus obtaining superintegrable ND spaces with non-constant curvature. Secondly, we present superintegrable ND generalizations of the maximally
superintegrable potentials in two dimensions introduced by Kalnins, Kress, Miller and Winternitz (J. Math. Phys. 43, (2002) 970; 44, (2003) 5811). Our
construction again relies on the sl(2)-coalgebra symmetry which appears in the analysis of some quadratically superintegrable systems. Furthermore, we
show that some of the Hamiltonians discussed by Kalnins et al. can actually be understood as intrinsic harmonic oscillator potentials on such ND
generalized
Darboux spaces.

Joint work with with A. Ballesteros, A. Enciso and O. Ragnisco