Completion of linear differential systems to involution

In this paper we generalize the involutive methods and algorithms have been devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutive algorithms. We present an algorithm for computation of a minimal involutive differential basis. Its correctness and termination hold for any constructive and noetherian involutive division. As two important applications we consider posing of an initial value problem for a linear differential system providing uniqueness of its solution and Lie symmetry analysis of nonlinear differential equations. In particular, this allows to determine the structure of arbitrariness in general solution of linear systems and thereby to find the size of symmetry group.