  We study conditions under which a partial differential operator of arbitrary
order $n$ in two variables or ordinary linear differential operator admits a
factorization with a first-order factor on the left. The factorization process
consists of solving, recursively, systems of linear equations, subject to
certain differential compatibility conditions. In the generic case of partial
differential operators one does not have to solve a differential equation. In
special degenerate cases, such as ordinary differential, the problem is finally
reduced to the solution of some Riccati equation(s). The conditions of
factorization are given explicitly for second- and, and an outline is given for
the higher-order case.
