We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\overline{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $\zeta$ of $\overline{M}$ is tangent to $M$. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\overline{M}$ such that (1) the screen distribution $S\left(TM\right)$ is totally umbilical or (2) $M$ is screen conformal.

A primary Hopf surface is a compact complex surface with universal cover ${ℂ}^2-\left\{\right(0,0\left)\right\}$ and cyclic fundamental group generated by the transformation $(u,v)\mapsto (\alpha u+\lambda v^m,\beta v)$, $m\in \mathbb{Z}$, and $\alpha ,\beta ,\lambda \in ℂ$ such that $\mid \alpha \mid \ge \mid \beta \mid \>1$ and $(\alpha -{\beta }^m)\lambda =0$. Being diffeomorphic with $S^3\times S^1$ Hopf surfaces cannot admit any Kähler metric. However, it was known that for $\lambda =0$ and $\mid \alpha \mid =\mid \beta \mid$ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class $1$ ($\lambda =0$). We also show...