On pseudo-Hermitian Einstein spaces

We describe and construct here pseudo-Hermitian structures $\theta$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $d\theta$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from K{\"a}hler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metrics of the pseudo-Hermitian Einstein structures. These Fefferman metrics are never Einstein, but they are locally always conformally Einstein.