The vacuum weighted Einstein field equations

On a spacetime $(M,g)$ endowed with a density function $h$, we consider the vacuum weighted Einstein field equations: $$h\rho -{\mathrm{Hes}}_h+\mathrm{\Delta }hg=0.$$ First, it is shown that the equation characterizes critical metrics for an appropriate action. Then, after describing locally conformally flat solutions in arbitrary dimension, four-dimensional solutions with harmonic curvature are classified.