η-parallel contact metric spaces

We prove that a contact metric manifold M=(M;η,ξ,φ,g) with η-parallel tensor h is either a K-contact space or a (k,μ)-space, where h denotes, up to a scaling factor, the Lie derivative of the structure tensor φ in the direction of the characteristic vector ξ. In the latter case, its associated CR-structure is in particular integrable.