  We consider structures analogous to symplectic Lefschetz pencils in the
context of a closed 4-manifold equipped with a `near-symplectic' structure (ie,
a closed 2-form which is symplectic outside a union of circles where it
vanishes transversely). Our main result asserts that, up to blowups, every
near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic
Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates
the boundaries of the Lefschetz fibrations to each other via a sequence of
fibrewise handle additions taking place in a neighbourhood of the zero set of
the 2-form. Conversely, from such a decomposition one can recover a
near-symplectic structure.
