Almost Kenmotsu manifolds with a condition of η-parallelism

We consider almost Kenmotsu manifolds (M²ⁿ⁺¹,φ,ξ,η,g) with η-parallel tensor h^′=h○φ, 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition _ξh^′=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M²ⁿ⁺¹ are introduced and studied.