  The physical and mathematical properties of the non-linearly coupled
black-hole-orbiting-ring system are studied analytically to second order in the
dimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of the
black-hole horizon (here $M_{\text{ir}}$ is the irreducible mass of the slowly
rotating central black hole). In particular, we determine analytically, to
first order in the dimensionless ring-to-black-hole mass ratio
$m/M_{\text{ir}}$, the shift $\Delta\Omega_{\text{mb}}/\Omega_{\text{mb}}$ in
the orbital frequency of the {\it marginally bound} circular geodesic that
characterizes the composed curved spacetime. Interestingly, our analytical
results for the frequency shift $\Delta\Omega_{\text{mb}}$ in the composed
black-hole-orbiting-ring toy model agree qualitatively with the recently
published numerical results for the corresponding frequency shift in the
physically related (and mathematically much more complex)
black-hole-orbiting-particle system. In particular, the present analysis
provides evidence that, at order $O(m/M_{\text{ir}})$, the recently observed
positive shift in the angular frequency of the marginally bound circular orbit
is directly related to the physically intriguing phenomenon of dragging of
inertial frames by orbiting masses in general relativity.
