  Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic
definition of the associated Goldman flow, which is a circle action on a dense
open subset of the moduli space of equivalence classes of flat
SU(2)-connections over the surface. A cylinder in a compact nonorientable
surface lifts to two cylinders in the orientable double cover, and the
"composite flow" is the composition of one of the associated flows with the
inverse flow of the other. Providing explicit descriptions, we relate the flow
on the moduli space of the nonorientable surface with the composite flow on the
moduli space of the double cover. We prove that the composite flow preserves a
certain Lagrangian submanifold.
