  We study the cohomology properties of the singular foliation $\F$ determined
by an action $\Phi \colon G \times M\to M$ where the abelian Lie group $G$
preserves a riemannian metric on the compact manifold $M$. More precisely, we
prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is
finite dimensional and verifies the Poincar\'e Duality. This duality includes
two well-known situations:
  -- Poincar\'e Duality for basic cohomology (the action $\Phi$ is almost
free).
  -- Poincar\'e Duality for intersection cohomology (the group $G$ is compact
and connected).
