Gradient Einstein solitons

In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon. We show that these structures are more rigid than standard Ricci solitons. It turns out that this property holds also in the Lorentzian setting and for a more general class of structures which includes some gravitational theories. We prove several classification results both in the compact and the noncompact case and we provide at the same time existence results for rotationally symmetric solutions.