The Orbit Space Approach to the Theory of Phase Transitions: The Non-Coregular Case

We consider the problem of the determination of the isotropy classes of the orbit spaces of all the real linear groups, with three independent basic invariants satisfying only one independent relation. The results are obtained in the $\hat P$-matrix approach solving a universal differential equation ({\em master equation}) which involves as free parameters only the degrees $d_a$ of the invariants. We begin with some remarks which show how the $\hat{P}$-matrix approach may be relevant in physical contexts where the study of invariant functions is important, like in the analysis of phase spaces and structural phase transitions (Landau's theory).