  We study properties of generalized frieze varieties for quivers associated to
cluster automorphisms. Special cases include acyclic quivers with Coxeter
automorphisms and quivers with Cluster DT automorphisms. We prove that the
generalized frieze variety X of an affine quiver with the Coxeter automorphism
is either a finite set of points or a union of finitely many rational curves.
In particular, if dim X=1, the genus of each irreducible component is zero. We
also propose an algorithm to obtain the defining polynomials for each
irreducible component of the generalized frieze variety for affine quivers.
Furthermore, we give the Gr\"obner basis with respect to a given monomial order
for each irreducible component of frieze varieties of affine quivers with given
orientations, and show that each component is a smooth rational curve.
