In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, https://en.wikipedia.org/wiki/Symbol_of_a_differential_operator). But there are easy examples for which this information is not enough. Consider the two-dimensional system: $$ (\partial+1)\phi(z,z')=0 $$ $$ (\partial+\partial')\phi(z,z')=0 $$ and, $$ (\partial+z)\psi(z,z')=0 $$ $$ (\partial+\partial')\psi(z,z')=0. $$ Both have the same principal symbols, namely, $p$ and $p+p'$, however, the first has the solution $\phi(z,z')=e^{z'-z}$, while the second has no solution (other than $\psi(z,z')=0$).
Clearly the two systems are distinct, and yet, it has been suggested that they both have one non-trivial solution within the context of the Kashiwara index theorem.
Any insight is greatly appreciated. I have a feeling it has to do with the fact that they are only related micro-locally, but I can't see how this relates the solution space.