  In this paper, we provide a systematic and constructive description of
Vaisman structures on certain principal elliptic bundles over complex flag
manifolds. From this description we explicitly classify homogeneous l.c.K.
structures on compact homogeneous Hermitian manifolds using elements of
representation theory of complex simple Lie algebras. Moreover, we also
describe using Lie theory all homogeneous solutions of the
Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weyl
manifolds. As an application, we provide a huge class of explicit (nontrivial)
examples of such structures on homogeneous Hermitian manifolds, these examples
include elliptic bundles over full flag manifolds, elliptic bundles over
Grassmannian manifolds, and 8-dimensional locally conformal hyperK\"{a}hler
compact manifolds.
