  Suppose that a three-manifold M contains infinitely many distinct strongly
irreducible Heegaard splittings H + nK, obtained by Haken summing the surface H
with n copies of the surface K. We show that K is incompressible. All known
examples, of manifolds containing infinitely many irreducible Heegaard
splittings, are of this form. We also give new examples of such manifolds.
