Homogeneous 4-dimensional Kaehler--Weyl Structures
  Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting. 
M. Brozos-Vazquez, E. Garcia-Rio, P. Gilkey, and R. Vazquez-Lorenzo
math.DG