Three-dimensional noncompact trans-Sasakian $η$-Einstein manifolds

We classify connected, simply connected, noncompact three-dimensional trans-Sasakian $η$-Einstein manifolds. In [1], we showed that, in the compact case, such a manifold is either of $β$-Kenmotsu type or homothetic to a Sasakian manifold. In this paper, in the noncompact case, we find families of trans-Sasakian $η$-Einstein manifolds that are neither $β$-Kenmotsu nor homothetic to Sasakian manifolds. These families have infinitely many degrees of freedom. We give these nontrivial metrics explicitly. We also give a structure theorem for the case in which the characteristic vector field is Palais-nonregular.