  We consider a curved Sitnikov problem, in which an infinitesimal particle
moves on a circle under the gravitational influence of two equal masses in
Keplerian motion within a plane perpendicular to that circle. There are two
equilibrium points, whose stability we are studying. We show that one of the
equilibrium points undergoes stability interchanges as the semi-major axis of
the Keplerian ellipses approaches the diameter of that circle. To derive this
result, we first formulate and prove a general theorem on stability
interchanges, and then we apply it to our model. The motivation for our model
resides with the $n$-body problem in spaces of constant curvature.
