  A quantum sl(2,R) coalgebra (with deformation parameter z) is shown to
underly the construction of superintegrable Kepler potentials on 3D spaces of
variable and constant curvature, that include the classical spherical,
hyperbolic and (anti-)de Sitter spaces as well as their non-constant curvature
analogues. In this context, the non-deformed limit z = 0 is identified with the
flat contraction leading to the proper Euclidean and Minkowskian
spaces/potentials. The corresponding Hamiltonians admit three constants of the
motion coming from the coalgebra structure. Furthermore, maximal
superintegrability of the Kepler potential on the spaces of constant curvature
is explicitly shown by finding an additional constant of the motion coming from
an additional symmetry that cannot be deduced from the quantum algebra. In this
way, the Laplace-Runge-Lenz vector for such spaces is deduced and its algebraic
properties are analysed.
