  We study the effective action for the massive vector field theory
non-minimally coupled to external gravitational field. Such a theory is an
interesting model both from the theoretical side and also due to the various
phenomenological applications to cosmology and astrophysics. The present work
pretends to initiate a systematic study of its properties at the quantum level,
by exploring free massive vector coupled to an external symmetric second-rank
tensor. Stueckelberg scalar field is used to restore the gauge invariance.
After that, by using a special gauge fixing and non-local in external fields
change of variables, we diagonalize the bilinear form of the action and develop
a consistent procedure to study the effective action. As a result we derive a
complete non-linear structure of divergences of the effective action and
discuss its properties.
