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 $a,b,c \in k$?

Thank you in advance.