  In this paper we analyze the Kerr geometry in the context of Conformal
Gravity, an alternative theory of gravitation, which is a direct extension of
General Relativity. Following previous studies in the literature, we introduce
an explicit expression of the Kerr metric in Conformal Gravity, which naturally
reduces to the standard General Relativity Kerr geometry in the absence of
Conformal Gravity effects. As in the standard case, we show that the
Hamilton-Jacobi equation governing geodesic motion in a space-time based on
this geometry is indeed separable and that a fourth constant of motion-similar
to Carter's constant-can also be introduced in Conformal Gravity. Consequently,
we derive the fundamental equations of geodesic motion and show that the
problem of solving these equations can be reduced to one of quadratures. In
particular, we study the resulting time-like geodesics in Conformal Gravity
Kerr geometry by numerically integrating the equations of motion for Earth
flyby trajectories of spacecraft. We then compare our results with the existing
data of the Flyby Anomaly in order to ascertain whether Conformal Gravity
corrections are possibly the origin of this gravitational anomaly. Although
Conformal Gravity slightly affects the trajectories of geodesic motion around a
rotating spherical object, we show that these corrections are minimal and are
not expected to be the origin of the Flyby Anomaly, unless conformal parameters
are drastically different from current estimates. Therefore, our results
confirm previous analyses, showing that modifications due to Conformal Gravity
are not likely to be detected at the Solar System level, but might affect
gravity at the galactic or cosmological scale.
