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.
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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. 



Example of non-holonomic D-module and explicit computation of characteristic variety
(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.)


