Orbit space reduction and localizations

We review the familiar method of reducing a symmetric ordinary differential equation via invariants of the symmetry group. Working exclusively with polynomial invariants is problematic: Generator systems of the polynomial invariant algebra, as well as generator systems for the ideal of their relations, may be prohibitively large, which makes reduction unfeasible. In the present paper we propose an alternative approach which starts from a characterization of common invariant sets of all vector fields with a given symmetry group, and uses suitably chosen localizations. We prove that there exists a reduction to an algebraic variety of codimension at most two in its ambient space. Some examples illustrate the approach.