I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration structure.

So I wonder if every symplectic manifold (especially symplectic 4-manifold) admits a Lefschetz fibration instead of just a Lefschetz pencil? If not, are there any examples?