  Black hole candidates in X-ray binary systems and at the centers of galaxies
are expected to be the Kerr black holes of General Relativity, but the actual
nature of these objects has yet to be verified. In this paper, we consider the
possibility that they are exotic compact objects and we describe their exterior
gravitational field with a subclass of the Manko-Novikov metrics, which are
exact solutions of the vacuum Einstein's equations and can describe the
spacetime geometry around bodies with arbitrary mass-multipole moments. We
point out that around a Manko-Novikov object there may exist many disconnected
non-plunging regions at small radii, with no counterpart in the Kerr
background, and that their existence may be tested. For instance, in the
presence of an accretion disk, they may be filled by the accreting gas, forming
a ring structure that might remind the rings of Saturn. We suggest that the
existence of these regions may have a clear observational signature in the
waveform of the gravitational radiation emitted by an EMRI: in the last stage
of the inspiral, the waveform would be the combination of "regular chirps",
produced when the small object orbits in one of the non-plunging regions, and
"bursts", released when the small object jumps from a non-plunging region to
another one at smaller radii. Our conclusions are supported by some numerical
calculations of trajectories in the geodesic approximation, in which a particle
plunges from the ISCO and then seems to get trapped in the potential well at
smaller radii.
