Use of complex Lie symmetries for differential equations

Abstract:
Lie's approach for solving real differential equations can be extended to the study of complex Lie symmetries and their use in solving complex differential equations. A complex ordinary differential equation is a combination of two real partial differential equations with the constraint of the Cauchy-Riemann equations which constitutes an over determined system. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of a given complex differential equation. Further use of a complex Lie symmetry reduces the order of a complex ordinary differential equation which in turn yields the reduction by two real valued functions.