  We study the case when a bivariate Linear Partial Differential Operator
(LPDO) of orders three or four has several different factorizations.
  We prove that a third-order bivariate LPDO has a first-order left and right
factors such that their symbols are co-prime if and only if the operator has a
factorization into three factors, the left one of which is exactly the initial
left factor and the right one is exactly the initial right factor. We show that
the condition that the symbols of the initial left and right factors are
co-prime is essential, and that the analogous statement "as it is" is not true
for LPDOs of order four.
  Then we consider completely reducible LPDOs, which are defined as an
intersection of principal ideals. Such operators may also be required to have
several different factorizations. Considering all possible cases, we ruled out
some of them from the consideration due to the first result of the paper. The
explicit formulae for the sufficient conditions for the complete reducibility
of an LPDO were found also.
