  This paper is devoted to the classification of GL^+(2,R)-orbit closures of
surfaces in the intersection of the Prym eigenform locus with various strata of
quadratic differentials. We show that the following dichotomy holds: an orbit
is either closed or dense in a connected component of the Prym eigenform locus.
  The proof uses several topological properties of Prym eigenforms, which are
proved by the authors in a previous work. In particular the tools and the proof
are independent of the recent results of Eskin-Mirzakhani-Mohammadi.
  As an application we obtain a finiteness result for the number of closed
GL^+(2,R)-orbits (not necessarily primitive) in the Prym eigenform locus
Prym_D(2,2) for any fixed D that is not a square.
