Let us consider two vector fields (1) X' = F(X) (2) Y' = F(Y) defined on a give Euclidean space E where F and G are of class CN+1. Furthermore, assume that there is a smooth compact manifold M smoothly imbedded in E and that M is invariant for both vector fields. Also that F and G agree on M, i.e. F|M = G|M. We wish to study the question of CS-conjugacies between (1) and (2).