  The gravitational field of a massive, filamentary ring is considered. We
provide an analytic expression for the gravitational potential and demonstrate
that the exact gravitational potential and its gradient, thus the gravitational
force-field, is not central. Hence it is a good candidate to discuss the
difference between the concepts of center of mass and center of gravity. We
focus on other consequences of reduced symmetry, e.g., only the $z$-component
of the angular momentum is conserved. However, the remnant high symmetry of
this system also ensures that there are special classes of motions which are
restricted to invariant subspaces, thus, depending on the initial condition,
the dynamics of a point particle is integrable. We also show that periodic
orbits in the equatorial plane external to the ring are possible, but only if
the angular momentum is above a threshold value. In this case the orbits are
stable.
