Fat modules on some algebras.

Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely presented $k$-algebra $A$ classification of fat module on $A$ is known. For example, it is know for the easiest case $$A=k\langle x,y,z \rangle /(xy=ayx,xz=bzx,yz=czy)$$ with $a,b,c \in k$?

Can the classification for the algebras you exhibit be found in Artin's paper "Geometry of quantum planes"? Also, I would call the modules you are interested in fat \emph{point modules}, although I don't know how standard this is - the word `fat' to me would mean that the module was cyclic with Hilbert series $\frac{e(M)}{(1-t)^n}$ with $e(M)>1$ with $n$ any natural number. –  Andrew Davies Nov 27 '12 at 9:00