Canonical forms for nonlinear systems

The Hunt-Su linearization theorem for single input systems utilizes the feedback group, a subgroup of Gl(2,R). The classical Goursat-Von Weber theorem may be viewed as a "linearization" theorem utilizing all of Gl(2,R). A first goal is to show the relationship between These two theorems. The feedback group preserves parameterization of solutions. If one is interested only in preserving orbits, one can extend the feedback group to the feedback/reparameterization (or F/R) group which contains the feedback group but is properly contained in Gl(2,R). Conditions for linearization via the F/R group are given.