ABSTRACT
It is assumed now that for a given ode of order not higher than three with nontrivial Lie symmetries the transformation functions from actual variables x and y(x) to canonical variables u and v(u) are known. This implies that its infinitesimal symmetry generators are explicitly known in canonical variables from the tabulation in Section 5.1 without further calculations. What remains to be done is to solve the canonical equation, and to generate the solution in actual variables from it. The term solution always means general solution; i.e., a n-parameter family of solutions if n is the order of the equation. This excludes the so-called singular solutions which may or may not be specializations of the general solution. The relation between these two concepts is discussed in detail in the article by Buium and Cassidy [23], Section 1.8, and articles by Ritt quoted there. Before this proceeding is described in detail for the individual symmetry
classes, the underlying general principles will be outlined. The following discussion is based on Engel [44], vol. V, Anmerkungen by Engel, pages 643-669 and 682-687 where many more details may be found. Let the canonical ode of order n have an r-parameter symmetry group.
The further proceeding depends crucially on the relative values of r and n. At first it is assumed that r < n and ∆ 6= 0; the Lie determinant ∆ is defined by (5.15). The fundamental invariants Φ1 and Φ2 defined on page 201 are of order r − 1 and r respectively. In terms of these invariants the canonical equation has the form
