in $S$, obtained by solving the system of linear equations $h\cdot x=x\ \forall h\in H$. The allowed range of $x$, $x\in{\mathcal V}$, is determined by the inequalities assuring that rank$(P(x))=$dim$(\widehat S)$. The parameters $x$, with range $\mathcal V$, would provide, in this case, a one-to-one parametrization of the stratum, thus solving our problem. In general, however, the intersection of a $G$-orbit, of orbit–type $(H)$, with the stratum $S$ does not reduce to a single point. So, a sounder analysis is required, that will be the object of the rest of this section[^2]. Two distinct points, $x$ and $g\cdot x$, $g\in G$, of the same $G$-orbit, lie in $\mathcal V$ iff $h\cdot g\cdot x = g\cdot x\
\forall h\in H$, that is, iff $g$ is in the stabilizer Stab$(H,
G)$ of $H$ in $G$. The intersection of a $G$-orbit of $S$ with ${\mathcal V}$ is, therefore, the Stab($H$, $G$)-orbit through $x$ and Stab($H$, $G$) is the largest subgroup of $G$ leaving $V$ invariant.

In the group space (Stab($H$, $G$),V), the isotropy subgroup at a point of general position is $H$. Therefore, the principal stratum $\Sigma$ satisfies the following relation:

[$${V\supset \Sigma\supseteq {\mathcal V}.}\label{ins}$$]{} It has to be noted that $\Sigma$ could 