  We investigate the relation between holomorphic torus actions on complex
manifolds of LCK type and the existence of special LCK metrics. We show that if
the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real
compact torus, then there exists a Vaisman metric on the manifold. Moreover, we
show that if the group of biholomorphisms contains a compact torus whose
dimension is half the real dimension of $M$, then $(M,J)$ admits an LCK metric
with positive potential. As an application, we obtain a classification of
manifolds of LCK type among all the manifolds having the structure of a
holomorphic principal torus bundle. Moreover, we obtain new non-existence
results for LCK metrics on certain products of complex manifolds.
