Let $S$ be a closed connected orientable surface with $g(S)>0$. Jennifer Schultens, in her paper ``The Classification of Heegaard Splittings for (Compact Orientable Surface)$\times S^1$'', proves that $S\times S^1$ does not admit any strongly irreducible Heegaard splitting. My questions are:

- Are there other (i.e. non-trivial) closed circle bundles which admit no strongly irreducible Heegaard splittings? Furthermore, is there a classification of closed circle bundles which admit no strongly irreducible Heegaard splittings?
- Are there other surface bundles which admit no strongly irreducible Heegaard splittings?