"Involution and Lie Symmetry Analysis of Differential Equations" Vladimir Gerdt Laboratory of Computing Techniques and Automation Joint Institute for Nuclear Research 141980 Dubna, Russia Email: gerdt@jinr.ru ABSTRACT: We present a general algorithmic approach to completion to involution of linear systems of partial differential equations. We consider an algorithm for computation of the minimal involutive system. An important application of the new algorithm is Lie symmetry analysis of nonlinear differential equations. For construction of Lie symmetry generators one needs to integrate their determining equations which form a system of linear partial differential equations. This step of Lie symmetry analysis is generally the most difficult one, and completion of the determining equations to involution is the most universal algorithmic tool of their integration. Another important application of the involutive method is posing of an initial value problem providing the unique solution of a system of partial differential equations. For linear involutive systems we formulate such a well-posed initial value problem and thereby generalize the classical results of Janet to arbitrary involutive divisions. This formulation makes it possible, among other things, to reveal the structure of arbitrariness in general solution. In particular, for determining systems occurring in Lie symmetry analysis it allows one to find the size of symmetry groups, that has been shown by F.Schwarz for Janet division.