  The McGehee regularization is a method to study the singularity at the origin
of the dynamical system describing a point particle in a plane moving under the
action of a power-law potential. It was used by Belbruno and Pretorius to
perform a dynamical system regularization of the singularity at the center of
the motion of massless test particles in the Schwarzschild spacetime. In this
paper, we generalize the McGehee transformation so that we can regularize the
singularity at the origin of the dynamical system describing the motion of
causal geodesics (timelike or null) in any stationary and spherically symmetric
spacetime of Kerr-Schild form. We first show that the geodesics for both
massive and massless particles can be described globally in the Kerr-Schild
spacetime as the motion of a Newtonian point particle in a suitable radial
potential and study the conditions under which the central singularity can be
regularized using an extension of the McGehee method. As an example, we apply
these results to causal geodesics in the Schwarzschild and Reissner-Nordstr\"om
spacetimes. Interestingly, the geodesic trajectories in the whole maximal
extension of both spacetimes can be described by a single two-dimensional phase
space with non-trivial topology. This topology arises from the presence of
excluded regions in the phase space determined by the condition that the
tangent vector of the geodesic be causal and future directed.
