  We study static cosmic string solutions in the context of Bekenstein-type
models. We show that there is a class of models of this type for which the
classical Nielsen-Olesen vortex is still a valid solution. However, in general
static string solutions in Bekenstein-type models strongly depart from the
standard Nielsen-Olesen solution with the electromagnetic energy concentrated
along the string core seeding spatial variations of the fine structure
constant, $\alpha$. We consider models with a generic gauge kinetic function
and show that equivalence principle constraints impose tight limits on the
allowed variations of $\alpha$ induced by string networks on cosmological
scales.
