ABSTRACT
It is assumed now that a differential equation of order two or three allows a nontrivial symmetry of a known type or, if its order is one, at least a single symmetry generator is explicitly known. This knowledge is utilized for transforming it into a canonical form corresponding to its symmetry class whenever this is possible. To this end a system of equations is constructed the solutions of which are the desired transformation functions. These equations must be such that algorithms are available for determining its solutions in well-defined function fields, e. g. rational or Liouvillian functions. A complete understanding of these equations is achieved by determining for each symmetry class the structure invariance group of its canonical form equations, and the differential invariants that may be constructed from the transformation functions.
