  We propose a classification of knots in S^1 x S^2 that admit a longitudinal
surgery to a lens space. Any lens space obtainable by longitudinal surgery on
some knots in S^1 x S^2 may be obtained from a Berge-Gabai knot in a Heegaard
solid torus of S^1 x S^2, as observed by Rasmussen. We show that there are yet
two other families of knots: those that lie on the fiber of a genus one fibered
knot and the `sporadic' knots. All these knots in S^1 x S^2 are both doubly
primitive and spherical braids.
  This classification arose from generalizing Berge's list of doubly primitive
knots in S^3, though we also examine how one might develop it using Lisca's
embeddings of the intersection lattices of rational homology balls bounded by
lens spaces as a guide. We conjecture that our knots constitute a complete list
of doubly primitive knots in S^1 x S^2 and reduce this conjecture to
classifying the homology classes of knots in lens spaces admitting a
longitudinal S^1 x S^2 surgery.
