For about twenty five years it was a kind of folk theorem that complex vector-fields defined on (with open set in ) by

with analytic, were subelliptic as soon as they were hypoelliptic. This was the case when , but in the case , an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that does not admit any local maximum or minimum (through a nonstandard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M. Trépreau show by examples that there are very simple systems (with polynomial 's) which are hypoelliptic but not subelliptic in the standard -sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when is ) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest nontrivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.