  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 there exist physically reasonable models
with arbitrarily many necks. We also show how a first order phase transition
can give rise to quite pronounced secondary double necks. The analysis is
carried out using a modification of a recent dynamical systems formulation of
the TOV equations due to Nilsson and Uggla. Our reformulation allows for a very
general family of equations of state including, for example, phase transitions.
