  We generalize the usual abelian Berry phase generated for example in a system
with two non-degenerate states to the case of a system with two doubly
degenerate energy eigenspaces. The parametric manifold describing the space of
states of the first case is formally given by the G(2,1) Grassmannian manifold,
while for the generalized system it is given by the G(4,2) one. For the latter
manifold which exhibits a much richer structure than its abelian counterpart we
calculate the connection components, the field strength and the associated
geometrical phases that evolve non-trivially both of the degenerate
eigenspaces. A simple atomic model is proposed for their physical
implementation.
