  Let $(A,\mathfrak{m})$ be a commutative complete equicharacteristic
Gorenstein isolated singularity of dimension $d $ with $k = A/\mathfrak{m}$
algebraically closed. Let $\Gamma(A)$ be the AR (Auslander-Reiten) quiver of
$A$. Let $\mathcal{P}$ be a property of maximal Cohen-Macaulay $A$-modules. We
show that some naturally defined properties $\mathcal{P}$ define a union of
connected components of $\Gamma(A)$. So in this case if there is a maximal
Cohen-Macaulay module satisfying $\mathcal{P}$ and if $A$ is not of finite
representation type then there exists a family $\{ M_n \}_{n \geq 1}$ of
maximal Cohen-Macaulay indecomposable modules satisfying $\mathcal{P}$ with
multiplicity $e(M_n) > n$. Let $\underline{\Gamma(A)}$ be the stable quiver. We
show that there are many symmetries in $\underline{\Gamma(A)}$. As an
application we show that if $(A,\mathfrak{m})$ is a two dimensional Gorenstein
isolated singularity with multiplicity $e(A) \geq 3$ then for all $n \geq 1$
there exists an indecomposable self-dual maximal Cohen-Macaulay $A$-module of
rank $n$.
