  In this paper we regularize the Kepler problem on $S^3$ in several different
ways. First, we perform a Moser-type regularization. Then, we adapt the
Ligon-Schaaf regularization to our problem. Finally, we show that the Moser
regularization and the Ligon-Schaaf map we obtained can be understood as the
composition of the corresponding maps for the Kepler problem in Euclidean space
and the gnomonic transformation.
