  In the recent paper "Mutation in triangulated categories and rigid
Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of
isolated singularities over which it is possible to classify the indecomposable
maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper
we present two new approaches to these examples. In the first approach we give
a relation with cluster categories. In the second approach we use Orlov's
result on the graded singularity category. We obtain some new results on the
singularity category of isolated singularities which may be interesting in
their own right.
