  In this paper we show that for a given 3-manifold and a given Heegaard
splitting there are finitely many preferred decomposing systems of $3g - 3$
disjoint essential disks. These are characterized by a combinatorial criterion
which is a slight strengthening of Casson-Gordon's rectangle condition. This is
in contrast to fact that in general there can exist infinitely many such
systems of disks which satisfy just the Casson-Gordon rectangle condition.
