  An LCK (locally conformally Kahler) manifold is a Hermitian manifold which
admits a Kahler cover with deck group acting by holomorphic homotheties with
respect to the Kahler metric. The product of two LCK manifolds does not have a
natural product LCK structure. It is conjectured that a product of two compact
complex manifolds is never LCK. We classify all known examples of compact LCK
manifolds onto three not exclusive classes: LCK with potential, a class of
manifolds we call of Inoue type, and those containing a rational curve. In the
present paper, we prove that a product of an LCK manifold and an LCK manifold
belonging to one of these three classes does not admit an LCK structure.
