  Given some type of fibration on a 4-manifold $X$ with a torus regular fiber
$T$, we may produce a new 4-manifold $X_T$ by performing torus surgery on $T$.
There is a natural way to extend the fibration to $X_T$, but a multiple fiber
(non-generic) singularity is introduced. We construct explicit generic
fibrations (with only indefinite fold singularities) in a neighborhood of this
multiple fiber. As an application this gives explicit constructions of broken
Lefschetz fibrations on all elliptic surfaces (e.g. the family $E(n)_{p,q}$).
As part of the construction we produce generic fibrations around exceptional
fibers of Seifert fibered spaces.
