Lie symmetries and CR geometry

This paper is divided into three parts. Part I develops a general, new theory (inspired by modern CR geometry) of Lie symmetries of completely integrable pde systems, viewed from their associated submanifolds of solutions. Part II constructs general combinatorial formulas for the prolongations of vector fields to jet spaces. Part III explicitly characterizes the flatness of some systems of the second order. The results presented here are original and were not published elsewhere; most formulas of Parts II and III were verified by means of Maple Release 7. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 47, Complex Analysis, 2007.