In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, http://www.sciencedirect.com/science/article/pii/S0001870802000993, and https://en.wikipedia.org/wiki/Symbol_of_a_differential_operator). In the first link, only the principal symbol is used to count the number of solutions to a holonomic system. But there are easy examples for which this information does not appear to be 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 $\psi(z,z')=0$).

It seems to be suggested in the first link that the second has a solution on an integral curve, perhaps, but it is not clear to me if that is the right context and if so, how does one construct such a curve?