  Zero modes of rotationally symmetric vortices in a hierarchy of generalized
Abelian Higgs models are studied. Under the finite-energy and the smoothness
condition, it is shown, that in all models, $n$ self-dual vortices superimposed
at the origin have $2n$ modes. The relevance of these modes for vortex
scattering is discussed, first in the context of the slow-motion approximation.
Then a corresponding Cauchy problem for an all head-on collision of $n$
vortices is formulated. It is shown that the solution of this Cauchy problem
has a $\frac{\pi}{n}$ symmetry.
