  We introduce the regularization Levi-Civita parameter for the `curved
Kepler', i.e., motion under the `Kepler-Coulomb' potential in a configuration
space with any constant curvature and metric of any signature type. Consistent
use of this parameter allows to solve the problem of motion (orbit shape and
time evolution along the orbit), thereby extending the use of the Levi-Civita
parameter beyond the usual Kepler problem in a flat Euclidean configuration
space. A `universal' description, where all relations are applicable to the
motions in any space and with any energy follow from our approach, with no need
to discuss separately the cases where the configuration space is flat or where
energy vanishes.
  We also discuss the connection of this `curved Kepler' problem with a
geodesic flow. The well known results by Moser, Osipov and Belbruno are shown
to hold essentially unchanged beyond the flat Euclidean configuration space.
`Curved' Kepler motions with a fixed value of the constant of motion
$\sigma:=-(2E - \kappa_1\kappa_2 J^2)$ on any curved configuration space with
constant curvature $\kappa_1$ and metric of signature type $\kappa_2$ can be
identified with the geodesic flow on a space with curvature $\sigma$ and metric
of the same signature type.
