Superstable theories with few countable models

Summary We prove here: Theorem. LetT be a countable complete superstable non -stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG ^eq orthogonal to the generics has Morley rank. Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.Supported by NSF grant DMS 90-06628