Let ?n (respectively, ??) be a free group of rank n (respectively, a free group of countable infinite rank). We consider the space of algebraic representations of the group ?n (respectively, ??) Hom(?n; PGL(2; C)) (respectively, Hom(??; PGL(2; C))). Inside each of these spaces we consider a couple of open and dense subsets. These subsets contain non-discrete groups of Möbius transformations. We proceed to find complex analytic parameters for these spaces given by fixed points.