  We describe a method to analyze causal geodesics in static and spherically
symmetric spacetimes of Kerr-Schild form which, in particular, allows for a
detailed study of the geodesics in the vicinity of the central singularity by
means of a regularization procedure based on a generalization of the McGehee
regularization for the motion of Newtonian point particles moving in a
power-law potential. The McGehee regularization was used by Belbruno and
Pretorius to perform a dynamical system regularization of the central
singularity of the motion of massless test particles in the Schwarzschild
spacetime. Our generalization allows us to consider causal (timelike or null)
geodesics in any static and spherically symmetric spacetime of Kerr-Schild
form. As an example, we apply these results to causal geodesics in the
Schwarzschild and Reissner-Nordstr\"om spacetimes.
