shall denote by $S$ the stratum of $(G,\real^n)$ with orbit–type $(H)$ and by $\widehat S$ its image $p(S)$ in the orbit space $\real^n/G$. Let us, also, define

$$V = \{x\in\real^n\mid G_x\supseteq H\},\qquad {\mathcal
V}=\{x\in\real^n \mid G_x = H\}\,. \label{1}$$

The non trivial sets $V$ and ${\mathcal V}$ have the following properties, which are more or less immediate consequences of their definitions:

1.  $V =\{x\in\real^n \mid h\cdot x=x, h\in H\}$ is a linear subspace of $\real^n$, let us call $\nu$ its dimensions;

2.  $V$ is the topological closure of $\mathcal V$: $V=\overline{\mathcal V}$;

3.  ${\mathcal V}=S \cap V $ and every $G$-orbit lying in $S$ has at least a point in ${\mathcal V}$ so that $S = G\cdot {\mathcal V}$, $\overline{S}=G\cdot V$ (where the bar denotes topological closure) and, consequently,

    [$${p({\mathcal V})={\widehat S},\qquad p(V)=\overline{\widehat
    S}}\label{pV}$$]{} and

    [$${{\rm rank}(P(x)\big|_{x\in{\mathcal V}})={\rm dim}(\widehat
    S),\qquad {\rm rank}(P(x)\big|_{x\in V\backslash{\mathcal
    V}})\,<\,{\rm dim}(\widehat S).}\label{rango}$$]{}

A $G$-orbit of $S$ may intersect $\mathcal V$ in one or more points. In the first case, every $G$-orbit of $S$ intersects $\mathcal V$ in a point and, owing to item [*(iii)*]{}, the coordinates of the points of ${\mathcal V}$ provide a one-to-one rational (in effect, linear) parametrizazion of the orbits of G lying 