  Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a
special subcomplex of the disk complex for one of the handlebodies of the
splitting, and then show that it is contractible. As applications, first we
show that the complex of Haken spheres for the splitting is contractible, which
refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus
two Heegaard splittings for non-prime 3-manifolds, which is a generalization of
the result of Montesinos-Safont. Finally, we show that the mapping class group
of the splitting, called the Goeritz group, is finitely presented by giving its
explicit presentation.
