  We consider the geometric properties of Hodge Cousin groups, introduced in an
unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups
corresponding to polarized $\mathbb{Q}$-Hodge structures. Basing on this
consideration, we introduce the class of abelian Cousin groups and prove an
analogue of Poincar\'e complete reducibility theorem for them.
