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In this paper, we provide a systematic and constructive description ofVaisman structures on certain principal elliptic bundles over complex flagmanifolds. From this description we explicitly classify homogeneous l.c.K.structures on compact homogeneous Hermitian manifolds using elements ofrepresentation theory of complex simple Lie algebras. Moreover, we alsodescribe using Lie theory all homogeneous solutions of theHermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weylmanifolds. As an application, we provide a huge class of explicit (nontrivial)examples of such structures on homogeneous Hermitian manifolds, these examplesinclude elliptic bundles over full flag manifolds, elliptic bundles overGrassmannian manifolds, and 8-dimensional locally conformal hyperK\"{a}hlercompact manifolds.

Principal elliptic bundles and compact homogeneous l.c.K. manifolds