  The system of four point vortices in the plane has relative equilibria that
behave as composite particles, in the case where three of the vortices have
strength $-\Gamma/3$ and one of the vortices has strength $\Gamma$. These
relative equilibria occur at nongeneric momenta. The reduction of this system,
at those momenta, by continuous and then discrete symmetries, classifies the
4-vortex states which have been observed as products of collisions of two such
composite particles. In this article I explicitly calculate these reductions,
and show they are qualitatively identical one degree of freedom systems on a
cylinder. The flows on these reduced systems all have one stable equilibrium
and one unstable equilibrium, and all the orbits are periodic except for two
homoclinic connections to the unstable equilibrium.
