In this talk, we outline an algorithm for factoring a linear homogeneous partial differential system whose coefficients are rational functions in two independent variables x, y, and whose solution space is finite-dimensional over the constant field. In other words, the algorithm computes all left ideals containing a given zero-dimensional left ideal in the ring of partial differential operators Dx and Dy over the field of rational functions in x and y.

Our approach is based on

1. a generalization of the associated equations technique used in the factorization of linear ode's, and
2. a method for finding hyperexponential solutions of zero-dimensional left ideals of differential operators.

We also introduce the notion of quotients, and discuss how to represent all the solutions of a given system by the solutions of its factors and quotients.