  We consider algorithms for the factorization of linear partial differential
operators. We introduce several new theoretical notions in order to simplify
such considerations. We define an obstacle and a ring of obstacles to
factorizations. We derive some interesting facts about the new objects, for
instance, that they are invariant under gauge transformations. An important
theorem for the construction of factoring algorithms is proved: a factorization
is defined uniquely from a certain moment on.
  For operators of orders three and two, obstacles are found explicitly.
