Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules when $R$ has finite type (ie. finitely many indecomposable MCM modules). For example, if $I$ is an ideal of $R$, then $End_R(I)$ can be identified with a subring of the integral closure of $R$ in its total quotient ring. In this case, it's not particularly difficult to compute $End_R(I)$ using well-known methods.

A particular example I would be interested in would be when $f = x^3 + y^4$. Here $R$ has two indecomposable MCM $R$-modules that are not isomorphic to ideals. I would be like to know if there are any explicit computations of the endomorphism ring for such modules $M$.

My motivation for doing so is to study the Quillen $K$-groups $K_1(mod \hspace{.125 cm} R)$ ($mod\hspace{.125 cm} R =$ finitely generated modules). In particular, I would like to study $Aut(M)$ and $Aut(M)_{ab}$. See this paper for details.