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The superposition of the Kepler-Coulomb potential on the 3D Euclidean spacewith three centrifugal terms has recently been shown to be maximallysuperintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] byfinding an additional (hidden) integral of motion which is quartic in themomenta. In this paper we present the generalization of this result to the NDspherical, hyperbolic and Euclidean spaces by making use of a unified symmetryapproach that makes use of the curvature parameter. The resulting Hamiltonian,formed by the (curved) Kepler-Coulomb potential together with N centrifugalterms, is shown to be endowed with (2N-1) functionally independent integrals ofthe motion: one of them is quartic and the remaining ones are quadratic. Thetransition from the proper Kepler-Coulomb potential, with its associatedquadratic Laplace-Runge-Lenz N-vector, to the generalized system is fullydescribed. The role of spherical, nonlinear (cubic), and coalgebra symmetriesin all these systems is highlighted.

Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces