  The aim of this paper is to classify three dimensional compact Riemannian
manifolds $(M^{3},g)$ that admits a non-constant solution to the equation
$$-\Delta f g+Hess f-fRic=\mu Ric+\lambda g,$$ for some special constants
$(\mu, \lambda)$, under assumption that the manifold has cyclic parallel Ricci
tensor. Namely, the structures that we will study here will be: positive static
triples, critical metrics of the volume functional, and critical metrics of the
total scalar curvature functional. We shall also classify $n$-dimensional
critical metrics of the volume functional with non-positive scalar curvature
and satisfying the cyclic parallel Ricci tensor condition.
