  In this paper we give a complete classification of minimal generating systems
in a very general class of Fuchsian groups G. This class includes for example
any G which has at least seven non-conjugate cyclic subgroups of order greater
than 2. In particular, the well known problematic cases where G has
characteristic exponents equal to 2 are not excluded. We classify generating
systems up to Nielsen equivalence; this notion is strongly related to Heegaard
splittings of 3-manifolds. The results of this paper provide in particular the
tools for a rather general extension of previous work of the authors and
others, on the isotopy classification of such splittings in Seifert fibered
spaces.
