A characterization of quasi-homogeneity in terms of liftable vector fields

We prove under certain conditions that any stable unfolding of a quasi-homogeneous map-germ with finite singularity type is substantial. We then prove that if an equidimensional map-germ is finitely determined, of corank 1, and either it admits a minimal stable unfolding or it is of multiplicity 3, then it admits a substantial unfolding if and only if it is quasi-homogeneous in some coordinate system. Based on this, we pose the following conjecture: a finitely determined map-germ is quasi-homogeneous in some coordinate system if and only if it admits a substantial unfolding.