  We consider the three body problem on $S^1$ under the cotangent potential. We
first construct homothetic orbits ending in singularities, including total
collision singularity and collision-antipodal singularity. Then certain
symmetrical periodic orbits with two equal masses, called Schubart orbits, are
shown to exist. The proof is based on the construction of a Wazewski set in the
phase space.
