Cohen-Macaulay modules over some non-reduced curve singularities

In this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]\mathrm{/}(xy,y^q-z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]\mathrm{/}(xy,z^2)$ we give an explicit description of all indecomposable Cohen–Macaulay modules and apply the obtained classification to construct explicit families of indecomposable matrix factorizations of $(xy{)}^2\in k[[x,y]]$.