  In this paper we introduce, in the Riemannian setting, the notion of
conformal Ricci soliton, which includes as particular cases Einstein manifolds,
conformal Einstein manifolds and (generic and gradient) Ricci solitons. We
provide here some necessary integrability conditions for the existence of these
structures that also recover, in the corresponding contexts, those already
known in the literature for conformally Einstein manifolds and for gradient
Ricci solitons. A crucial tool in our analysis is the construction of some
appropriate and highly nontrivial $(0,3)$-tensors related to the geometric
structures, that in the special case of gradient Ricci solitons become the
celebrated tensor $D$ recently introduced by Cao and Chen. A significant part
of our investigation, which has independent interest, is the derivation of a
number of commutation rules for covariant derivatives (of functions and
tensors) and of transformation laws of some geometric objects under a conformal
change of the underlying metric.
