  Even if the Fourier symbols of two constant rank differential operators have
the same nullspace for each non-trivial phase space variable, the nullspaces of
those differential operators might differ by an infinite dimensional space.
Under the natural condition of constant rank over $\mathbb{C}$, we establish
that the equality of nullspaces on the Fourier symbol level already implies the
equality of the nullspaces of the differential operators in $\mathscr{D}'$
modulo polynomials of a fixed degree. In particular, this condition allows to
speak of natural annihilators within the framework of complexes of differential
operators. As an application, we establish a Poincar\'{e}-type lemma for
differential operators of constant complex rank in two dimensions.
