We consider closed manifolds with $(Z_2)^r$-actions, which are obtained as intersections of products of spheres of a fixed dimension with certain 'generic' hyperplanes. The equivariant cohomology of these manifolds shows very interesting features. In particular one can realize this way all syzygies, which are in concordance with the general restriction found for the 'syzygy orders' of the equivariant cohomology of closed $(Z_2)^r$-manifolds.