Given a right and a left factors of a
Linear Partial Differential Operator, under which conditions we can
combine these two factorizations into one factorization, the product
of three factors? Numerous examples show that different situations
are possible, for example, operator
\[
L= (D_x D_y + 1 ) \circ (D_x + 1) = (D_x + 1) \circ (D_x D_y + 1 ) \
\]
has two factorizations, while its second-order factor $D_x D_y +1$
has no factorization at all. On the other hand, the existence of two
factorizations
\[
(D_x + x) \circ (D_{xy}+ yD_x +y^2 D_y + y^3)=A=(D_{xx}+ (x+y^2)D_x
+ xy^2) \circ (D_y + y)
\]
implies the existence of the ``complete'' factorization,
\[
A=(D_x + x) \circ (D_x +y^2) \circ (D_y +y) \ .
\]
In this talk we report on the obtained solution of this problem for
Linear Partial Differential Operator of arbitrary order and
depending on arbitrary number of variables. We also present some
further generalizations of the obtained results.