  Detailed numerical analyses of the orbital motion of a test particle around a
spinning primary are performed. They aim to investigate the possibility of
using the post-Keplerian (pK) corrections to the orbiter's periods (draconitic,
anomalistic and sidereal) as a further opportunity to perform new tests of
post-Newtonian (pN) gravity. As a specific scenario, the S-stars orbiting the
Massive Black Hole (MBH) supposedly lurking in Sgr A$^\ast$ at the center of
the Galaxy is adopted. We, first, study the effects of the pK Schwarzchild,
Lense-Thirring and quadrupole moment accelerations experienced by a target star
for various possible initial orbital configurations. It turns out that the
results of the numerical simulations are consistent with the analytical ones in
the small eccentricity approximation for which almost all the latter ones were
derived. For highly elliptical orbits, the size of all the three pK corrections
considered turn out to increase remarkably. The periods of the observed S2 and
S0-102 stars as functions of the MBH's spin axis orientation are considered as
well. The pK accelerations considered lead to corrections of the orbital
periods of the order of 1-100d (Schwarzschild), 0.1-10h (Lense-Thirring) and
1-10^3s (quadrupole) for a target star with a=300-800~AU and e ~ 0.8, which
could be possibly measurable by the future facilities.
