Cyclic pursuit systems provide a means to generate useful global behaviors in a collective of autonomous agents based on dyadic pursuit interactions between neighboring agents in a cycle graph. Here we consider a modified version of the cyclic pursuit framework in which a stationary beacon provides an additional reference for the agents in the system. Building on the framework proposed in our previous work, we derive necessary conditions for stability of circling equilibria in the n-agent system. Furthermore, we employ a change of variables to reveal the existence of a family of invariant manifolds related to spiral motions which maintain the formation shape up to geometric similarity.