reduction of differential equations by lie algebra of symmetries

The paper is devoted to an application of lie group theory to differential equations. the basic infinitesimal method for calculating symmetry group is presented, and used to determine general symmetry group of some differential equations. we include a number of important applications including integration of ordinary differential equations and finding some solutions of partial differential equations together with some examples. a bianchi theorem for the solvable symmetry groups is given to reduce a system of ordinary differential equations.