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The Rayleigh criterion is used to study the stability of circular orbits ofparticles moving around static black holes surrounded by different axiallysymmetric structures with reflection symmetry, like disks, rings and halos. Weconsider three models of disks one of infinite extension and two finite, andone model of rings. The halos are represented by external quadrupole moments(either oblate or prolate). Internal quadrupole perturbation (oblate andprolate) are also considered. For this class of disks the counter-rotationhypothesis implies that the stability of the disks is equivalent to thestability of test particles. The stability of Newtonian systems is alsoconsidered and compared with the equivalent relativistic situation. We findthat the general relativistic dynamics favors the formation of rings.Comment: RevTex, 9 pages, 14 eps figs. Phys. Rev. D to appea

Stability of circular orbits of particles moving around black holes surrounded by axially symmetric structures