CR submanifolds of maximal CR dimension in a complex Hopf manifold

We study CR submanifolds $M$ in a Hopf manifold $(\mathbb{CH}^N(\lambda), J_0,g_0)$ 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 $H(M)^\perp$ 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 $M$ fibers 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 $S\subset \mathbb{C}P^{N-1}$, for some geodesic hypersphere $S$, and (2) if $a_\xi (U)=0$ and $\mathrm{Spec}(a_\xi)=\{ 0, c \}$, for some $c \in \mathbb{R} \setminus \{ 0 \}$, then $M$ is locally a Riemannian product of totally geodesic submanifolds.