Invariants of the Rotation Group

In Section 3.5 we stated the conjecture that the center variety of family (3.3), or equivalently of family (3.69), always contains the variety V(Isym) as a component. This variety V(Isym) always contains the set R that corresponds to the time-reversible systems within family (3.3) or (3.69), which, when they arise through the complexiӿcation of a real family (3.2), generalize systems that have a line of symmetry passing through the origin. In Section 3.5 we had left incomplete a full characterization of R. To derive it we are led to a development of some aspects of the theory of invariants of complex systems of differential equations. Using this theory, we will complete the characterization of R and show that V(Isym) is actually its Zariski closure, the smallest variety that contains it. In the ӿnal section we will also apply the theory of invariants to derive a sharp bound on the number of axes of symmetry of a real planar system of differential equations.