Some aspects of phase transitions can be more conveniently studied in the orbit
space of the action of the symmetry group. After a brief review of the fundamental ideas
of this approach, I shall concentrate on the mathematical aspect and more exactly on
the determination of the equations dening the orbit space and its strata. I shall deal
only with compact coregular linear groups. The method exposed has been worked out
together with prof. G. Sartori and it is based on the solution of a matrix dierential
equation. Such equation is easily solved if an integrity basis of the group is known. If
the integrity basis is unknown one may determine anyway for which degrees of the basic
invariants there are solutions to the equation, and in all these cases also nd out the
explicit form of the solutions. The solutions determine completely the stratication of
the orbit spaces. Such calculations have been carried out for 2, 3 and 4-dimensonal orbit
spaces. The method is of general validity but the complexity of the calculations rises
tremendously with the dimension q of the orbit space. Some induction rules have been
found as well. They allow to determine easily most of the solutions for the (q + 1)-
dimensional case once the solutions for the q-dimensional case are known. The method
exposed is interesting because it allows to determine the orbit spaces without using any
specic knowledge of group structure and integrity basis and evidences a certain hidden
and yet unknown link with group theory and invariant theory.
