φ-symmetric contact metric spaces

We give the first examples of contact metric spaces which are weakly locally $\varphi$-symmetric (that is, $\nabla R=0$ on horizontal vectors), but not strongly (that is, not all reflections with respect to the characteristic lines are local isometries). These examples are three-dimensional non-unimodular Lie groups with a left-invariant contact metric structure. We exhibit additional symmetries on these spaces. 1991 Mathematics Subject Classification: 53B20, 53C15, 53C25.