  It is known that there are surface bundles of arbitrarily high genus which
have genus two Heegaard splittings. The simplest examples are Seifert fibered
spaces with the sphere as a base space, three exceptional fibers and which
allow horizontal surfaces. We characterize the monodromy maps of all surface
bundles with genus two Heegaard splittings and show that each is the result of
integral Dehn surgery in one of these Seifert fibered spaces along loops where
the Heegaard surface intersects a horizontal surface. (This type of surgery
preserves both the bundle structure and the Heegaard splitting.)
