Bach and Einstein’s equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization the classic Bach field equations on four-dimensional smooth manifold [Formula: see text] in presence text], given by map with source target another Riemannian manifold. Those are characterized vanishing two times covariant, symmetric, traceless conformally invariant tensor field, called text]-Bach tensor, that absence reduces related bi-energy text]. Since solutions Einstein-massless scalar equations, or more generally, Einstein wave solves those generalized Bach’s we include latter our analysis providing systematic study for them, relying recent concept text]-curvatures. We take opportunity discuss topic warped product solutions.