  Cohen-Macaulay Auslander algebras are the endomorphism algebras of
representation generators of the subcategory of Gorenstein projective modules
over $\rm{CM}$-finite algebras. In this paper, we study Cohen-Macaulay
Auslander algebras over $1$-Gorenstein algebras and
$\Omega_{\mathcal{G}}$-algebras. $1$-Gorenstein algebras are those of algebras
with global Gorenstein projective dimension at most one and
$\Omega_{\mathcal{G}}$-algebras are a class of algebras introduced in this
paper, including some important class of algebras for example Gentle algebras
and more generally quadratic monomial algebras. It will be shown how the
results for Gorenstein projective representations of a quiver over an Artin
algebra, including the submodule category introduced in [RS], or more
generally, the (separated) monomorphism category defined in [LZh2] and [XZZ],
can be applied to study the Cohen-Macaulay Auslander algebras.
