  We study the necessary and sufficient conditions on Abelianizable first class
constraints. The necessary condition is derived from topological considerations
on the structure of gauge group. The sufficient condition is obtained by
applying the theorem of implicit function in calculus and studying the local
structure of gauge orbits. Since the sufficient condition is necessary for
existence of proper gauge fixing conditions, we conclude that in the case of a
finite set of non-Abelianizable first class constraints, the Faddeev-Popov
determinant is vanishing for any choice of subsidiary constraints. This result
is explicitly examined for SO(3) gauge invariant model.
