  The Coulomb problem for Schr\"{o}dinger equation is examined, in spaces of
constant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base of
generalized parabolic coordinates. In contrast to the hyperbolic case, in
spherical space S_{3} such parabolic coordinates turn to be complex-valued,
with additional constraint on them. The technique of the use of such real and
complex coordinates in two space models within the method of separation of
variables in Schr\"{o}dinger equation with Kepler potential is developed in
detail; the energy spectra and corresponding wave functions for bound states
have been constructed in explicit form for both spaces; connections with
Runge-Lenz operators in both curved space models are described.
