  We give a complete description of all locally conformally K\"ahler structures
with holomorphic Lee vector field on a compact complex manifold of Vaisman
type. This provides in particular examples of such structures whose Lee vector
field is not homothetic to the Lee vector field of a Vaisman structure. More
generally, dropping the condition of being of Vaisman type, we show that on a
compact complex manifold, any lcK metric with potential and with holomorphic
Lee vector field admits a potential which is positive and invariant along the
anti-Lee vector field.
