  We study analytically a black-hole-ring system which is composed of a
stationary axisymmetric ring of particles in orbit around a perturbed Kerr
black hole of mass $M$. In particular, we calculate the shift in the orbital
frequency of the innermost stable circular orbit (ISCO) due to the finite mass
$m$ of the orbiting ring. It is shown that for thin rings of half-thickness
$r\ll M$, the dominant finite-mass correction to the characteristic ISCO
frequency stems from the self-gravitational potential energy of the ring (a
term in the energy budget of the system which is quadratic in the mass $m$ of
the ring). This dominant correction to the ISCO frequency is of order
$O(\mu\ln(M/r))$, where $\mu\equiv m/M$ is the dimensionless mass of the ring.
We show that the ISCO frequency increases (as compared to the ISCO frequency of
an orbiting test-ring) due to the finite-mass effects of the self-gravitating
ring.
