Factorization of overdetermined systems of linear partial differential equations with finite-dimensional solution space

We give an improved algorithm for factorization, i.e. decomposition of such systems with rational function coefficients into lower-order systems. This algorithm is based on the recent algorithm of Z. Li and F. Schwarz for construction of hyperexponential solutions. An analogue of the Loewy-Ore theory of factorization is exposed. We also prove an analogue of a result by E. Landau on uniqueness of the number and orders of irreducible factors.