CR submanifolds of maximal CR dimension in a complex Hopf manifold

We study CR submanifolds M in a Hopf manifold (CHN(lambda), J(0), g(0)) with the Boothby metric g(0), of maximal CR dimension. Any such M is a CR manifold of hypersurface type, although embedded in higher codimension, and its anti-invariant distribution H(M)(perpendicular to) is spanned by a unit vector field U. We classify the CR submanifolds M for which xi = -J(0)U is parallel in the normal bundle under assumptions on the spectrum of the Weingarten operator a(xi). We show that (1) if a(xi)(U) = (1/2)A (where A is the anti-Lee vector) and M fibres in tori over a CR submanifold of the complex projective space, then M lies on the (total space of the) pullback of the Hopf fibration via S subset of CPN-1, for some geodesic hypersphere S, and (2) if a(xi)(U) = 0 and Spec(a(xi)) = {0, c}, for some c is an element of R \ {0}, then M is locally a Riemannian product of totally geodesic submanifolds.