  We consider test particle motion in a gravitational field generated by a
homogeneous circular ring placed in $n$-dimensional Euclidean space. We observe
that there exist no stable stationary orbits in $n=6, 7, \ldots, 10$ but exist
in $n=3, 4, 5$ and clarify the regions in which they appear. In $n=3$, we show
that the separation of variables of the Hamilton-Jacobi equation does not occur
though we find no signs of chaos for stable bound orbits. Since the system is
integrable in $n=4$, no chaos appears. In $n=5$, we find some chaotic stable
bound orbits. Therefore, this system is nonintegrable at least in $n=5$ and
suggests that the timelike geodesic system in the corresponding black ring
spacetimes is nonintegrable.
