  In earlier work, Barchini, Kable, and Zierau constructed a number of
conformally invariant systems of differential operators associated to
Heisenberg parabolic subalgebras in simple Lie algebras. The construction was
systematic, but the existence of such a system was left open in two cases,
namely, the $\Omega_3$ system for type $A_2$ and type $D_4$. Here, such a
system is shown to exist for both cases. The construction of the system may
also be interpreted as giving an explicit homomorphism between generalized
Verma modules.
