Constructing stellar objects with multiple necks

We discuss the construction of perfect fluid stellar objects having optical geometries with multiple necks corresponding to spatially closed unstable lightlike geodesics. We prove that physically reasonable equations of state can give rise to stellar equilibria with arbitrarily many necks. We also s how how a first-order phase transition can give rise to quite pronounced se condary necks. The analysis is carried out using a modification of a recent dynamical systems formulation of the TOV equations due to Nilsson and Uggl a. Our reformulation allows for a very general family of equations of state including, for example, phase transitions.