Loewy decomposition of third-order linear aPDE's in the plane

Loewy's decomposition of a linear ordinary differential operator as the product of largest completely reducible components is generalized to partial differential operators of order three in two variables. This is made possible by considering the problem in the ring of partial differential operators where both left intersections and right divisors of left ideals are not necessarily principal. Listings of possible decomposition types are given. Many of them are illustraded by worked out examples. Algorithmic questions and questions of uniqueness are discussed in the Summary.