We first give a characterization for the set of real analytic diffeomorphisms which transform homogeneous vector fields of certain degree into homogeneous fields of the same degree with respect to an arbitrary dilation δεr. Such a set is constituted by the invertible analytic maps that are homogeneous of degree one with respect to δεr and can be endowed with the structure of a Lie Group whose Lie algebra is the space H1,r(ℝn) of the homogeneous fields of degree one with respect to δεr. Then we prove a decomposition theorem for the elements of the non semisimple Lie algebra H1,r(ℝn). This result is a non linear analog of the Jordan decomposition of a linear field, i.e. for X ∈ H1,r(ℝn), we can write X = S + N, with S linear semisimple and [S, N] = 0. We also give an explicit representation formula for the flow generated by a field in H1,r(ℝn). Finally we apply this result to obtain a simple representation for the trajectories of a class of affine control systems x˙=X0(x)+Bu, with X0 ∈ H1,r(ℝn) and В a constant field, that constitute a natural extension of the linear control systems.