Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

Abstract

Five equivalence classes had been found for systems of two second‐order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An “optimal (or simplest) canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5‐, 6‐, 7‐, 8‐, and 15‐dimensional Lie algebras. For those systems that arise from a scalar complex second‐order ordinary differential equation, treated as a pair of real ordinary differential equations, we provide a “reduced optimal canonical form.” This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6‐, 7‐, and 15‐dimensional algebras for these systems and illustrate our results with examples.