We briefly report on some recent and new results in geometric invariant theory, concerning the determination and classification of orbit spaces of compact linear groups. The results are obtained through the integration of a universal differential equation, which, for every compact linear group G, only requires as input the degrees of the elements of an integrity basis of the ideal of polynomial invariants of G. For coregular compact linear groups, all the orbit spaces, up to dimension 4, have been determined in this way. For higher dimensions there should be no difficulties, in principle. Our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking at the classical level.
