An example of a locally unsolvable differential equation of quasiprincipal type with a real-valued weighted principal symbol

In this note we show that the equation
$$ -\Bigl\{\Bigl(\frac1i\frac\partial{\partial x_1}\Bigr)^3+\Bigl(\frac1i\frac\partial{\partial x_2}\Bigr)^2+6i\Bigl(\frac1i\frac\partial{\partial x_1}\Bigr)\Bigl(\frac1i\frac\partial{\partial x_2}\Bigr)x_1\Bigr\}u=f $$
is locally unsolvable at the origin of the coordinate system. This equation belongs to the class that generalizes the principal type to the case of weighted derivatives. The example is interesting because the weighted principal symbol is real (in this situation, equations of principal type are solvable) but the unsolvability depends on the behavior of the lower-order terms in a neighborhood of the zeros of the weighted principal symbol.