  In this note we take a dynamical systems approach to the equations of motion
of a free test particle moving around a spherically symmetric static dilaton
black hole, written in the Einstein frame. The equations of motion are obtained
using the Euler-Lagrange formalism. Using the first integrals of motion, we
reach the conclusion that the free test particles are moving in a plane, named
\emph{plane of motion}. In it we analyze the existence and nature of the
equilibrium points and compare the behavior of free test particles near the
equilibrium points using the dynamics systems approach. The study revealed that
in the exact phase-plane exist distinct regions of motion, separated through a
curve named \emph{separatrix}. In the end we obtained a relation between the
parameters describing the black hole and the free test particle that holds on a
parabolic separatrix.
