Einstein condition and twistor spaces of compatible partially complex structures

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric. A generalization of the Einstein condition (discussed in the Besse book [Enstein Manifolds, Ergeb. Math. Grensgeb. (3), vol. 10, Springer, New York, 1987]) is also considered.