  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 defining 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 differential 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 find out the explicit form of the solutions. The solutions determine
completely the stratification 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 specific knowledge of group structure and
integrity basis and evidences a certain hidden and yet unknown link with group
theory and invariant theory.
