  We study the orbits in a Manko-Novikov type metric (MN) which is a perturbed
Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are
found in configuration space and on a surface of section for various values of
the energy E and the z-component of the angular momentum Lz. For relatively
large Lz there are two permissible regions of non-plunging motion bounded by
two closed curves of zero velocity (CZV), while in the Kerr metric there is
only one closed CZV of non-plunging motion. The inner permissible region of the
MN metric contains mainly chaotic orbits, but it contains also a large island
of stability. We find the positions of the main periodic orbits as functions of
Lz and E, and their bifurcations. Around the main periodic orbit of the outer
region there are islands of stability that do not appear in the Kerr metric. In
a realistic binary system, because of the gravitational radiation, the energy E
and the angular momentum Lz of an inspiraling compact object decrease and
therefore the orbit of the object is non-geodesic. In fact in an EMRI system
the energy E and the angular momentum Lz decrease adiabatically and therefore
the motion of the inspiraling object is characterized by the fundamental
frequencies which are drifting slowly in time. In the Kerr metric the ratio of
the fundamental frequencies changes strictly monotonically in time. However, in
the MN metric when an orbit is trapped inside an island the ratio of the
fundamental frequencies remains constant for some time. Hence, if such a
phenomenon is observed this will indicate that the system is non integrable and
therefore the central object is not a Kerr black hole.
