is an order parameter and $G$ is the symmetry group of the potential $\Phi(\alpha; x)$ (free energy, or Higgs potential, for instance), expressed in terms, also, of parameters $\alpha$. The points $x_0(\alpha)$, where the function $\phi_\alpha(x) =
\Phi(\alpha; x)$ takes on its absolute minimum, determine the stable phase of the system, whose residual symmetry is defined by the isotropy subgroup of $G$ at $x_0$. Owing to the $G$-invariance of the potential, each of its stationary points is degenerate along the $G$-orbit through it. Since the isotropy subgroups of $G$ at points of the same orbit are conjugate in $G$, only the orbit–type of $x_0(\alpha)$ is physically relevant. [*Structural phase transitions*]{} take place when, by varying the values of the $\alpha$’s, the point $x_0(\alpha)$ is shifted to a different stratum.

If $\Phi(\alpha;x)$ is a sufficiently general function of the $\alpha$’s, by varying these parameters, the point $x_0(\alpha)$ can be shifted to any stratum of $\real^n/G$. So, [*the strata are in a one-to-one correspondence with the symmetry phases allowed by the $G$-invariance of the potential*]{}. On the contrary, extra restrictions on the form of the potential function, not coming from G-symmetry requirements (e.g., the assumption that the potential is a polynomial of 