TITLE: Example of non-holonomic D-module and explicit computation of characteristic variety
QUESTION [6 upvotes]: I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know that for a system of differential equations $P$, the holonomicity of the $D$-module $D_X / D_X P$ means that the system $P$ is maximaly overdetermined. In order to see that, we compute the characteristic variety $\text{char}(D_X/D_X P)$ and if it is lagrangian in $T^* X$, the $D$-module is holonomic by definition. 
I'd like to compute explicitely the charactistic variety of two "easy" coherent $D$-modules in order to see the holonomicity. So to be precise, my question is 

Find two systems of differential equations $P$ and $Q$ on, let's say $\mathbb{C}^2$, such that $\text{char}(D_X/D_X P)$ and $\text{char}(D_X/D_X Q)$ can be explicitely computed in a short time and such that $\text{char}(D_X/D_X P)$ is lagrangian and $\text{char}(D_X/D_X Q)$ is not.

For example, I've also studied the notion of regularity and for that concept, the examples are easy : $z\partial_z -z$ is regular and $z\partial_z -1$ is not. But I couldn't find such easy examples for holonomicity. Perhaps it is linked with the complexity of the computation of characteristic varieties. (I know it is linked to Gröbner basis, but I don't know very well this theory)
Any examples or literature recommendations will be highly appreciated.

REPLY [8 votes]: (These examples are shamelessly pilfered from Gröbner Deformations of Hypergeometric Differential Equations by Saito, Sturmfels and Takayama, which is perhaps the place to learn computational D-module stuff.)
Holonomic: $D\cdot\{z_1\partial_2,z_2\partial_1\}$. This left ideal has  characteristic ideal of dimension 2, so it is holonomic.
Non-holonomic: let $f=(z_1^3-z_2^2)$; $D\cdot\{f\partial_1+\partial f/\partial z_1, f\partial_2+\partial f/\partial z_2\}=D\cdot\{\partial_1 f,\partial_2 f\}$. This left ideal has char.ideal of dimension 3, so it isn't holonomic.
(Edit: if you want to actually compute stuff, I recommend the Dmodules package for Macaulay2; it has tools to do most D-module things, including characteristic variety/ideal, gröbner bases etc.)