We discuss the Abelian vortex dynamics in the Abelian projection appro
ach to non-Abelian spin models. We show numerically that in the three-
dimensional SU(2) spin model in the maximal Abelian projection the Abe
lian off-diagonal vortices, unlike the diagonal vortices, are not resp
onsible for the phase transition. A generalization of the Abelian proj
ection approach to SU(N) spin models is briefly discussed. (C) 1998 Am
erican Institute of Physics. [S0021-3640(98)00208-4]