  We discuss the Morse-Novikov cohomology of a compact manifold, associated to
a closed one--form whose free abelian group generated by its periods $\langle
\int_\gamma \eta \mid [\gamma] \in \pi_1(M)\rangle$ is of rank 1, the focus
being on locally conformally symplectic manifolds. In particular, we provide an
explicit computation for the Inoue surface $\mathcal{S}^0$.
