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Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - I: Ordinary Differential Equations

The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of two real ordinary differential equations. The transformations that map a system of two nonlinear ordinary differential equations into systems of linear ordinary differential equations are obtained from complex transformations. Invariant criteria for linearization are given for second order complex ordinary differential equations in terms of the coefficients of the equations, as well as the corresponding real system, which provide procedures for writing down the solutions of the equations. Illustrative examples are given and discussed.