Use of complex Lie symmetries for differential equations
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.