  The Rayleigh criterion is used to study the stability of circular orbits of
particles moving around static black holes surrounded by different axially
symmetric structures with reflection symmetry, like disks, rings and halos. We
consider three models of disks one of infinite extension and two finite, and
one model of rings. The halos are represented by external quadrupole moments
(either oblate or prolate). Internal quadrupole perturbation (oblate and
prolate) are also considered. For this class of disks the counter-rotation
hypothesis implies that the stability of the disks is equivalent to the
stability of test particles. The stability of Newtonian systems is also
considered and compared with the equivalent relativistic situation. We find
that the general relativistic dynamics favors the formation of rings.
