The Schubart orbits on the circle

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.