  We show that generalized broken fibrations in arbitrary dimensions admit
rank-2 Poisson structures compatible with the fibration structure. After
extending the notion of wrinkled fibration to dimension 6 we prove that these
wrinkled fibrations also admit compatible rank-2 Poisson structures. In the
cases with indefinite singularities we can provide these wrinkled fibrations in
dimension 6 with near-symplectic structures.
