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.