  We study {\it non-holonomic} overideals of a left differential ideal
$J\subset F[\partial_x, \partial_y]$ in two variables where $F$ is a
differentially closed field of characteristic zero. The main result states that
a principal ideal $J=< P>$ generated by an operator $P$ with a separable {\it
symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a
finite number of maximal non-holonomic overideals. This statement is extended
to non-holonomic ideals $J$ with a separable symbol. As an application we show
that in case of a second-order operator $P$ the ideal $<P>$ has an infinite
number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In
case of a third-order operator $P$ we give few sufficient conditions on $<P>$
to have a finite number of maximal non-holonomic overideals.
