Decomposition of constraint systems is a key component of geometric constraint solving in CAD. On the other hand, some authors have introduced the notion of reparameterization which aims at helping the solving of indecomposable systems by replacing some geometric constraints by other ones. In previous works, the minimal change of the initial system is a main criterion. We propose to marry these two ingredients, decomposition and reparameterization, in a method able to reparameterize and to decompose a constraint system according to this reparameterization. As a result, we do not aim at minimizing the number of added constraints during the reparameterization, but we want to decompose the system such that each component owns a minimal number of such added constraints.