  We analyze the structure of the set of limiting Carleman weights in all
conformally flat manifolds, 3-manifolds, and 4-manifolds. In particular we give
a new proof of the classification of Euclidean limiting Carleman weights, and
show that there are only three basic such weights up to the action of the
conformal group. In dimension three we show that if the manifold is not
conformally flat, there could be one or two limiting Carleman weights. We also
characterize the metrics that have more than one limiting Carleman weight. In
dimension four we obtain a complete spectrum of examples according to the
structure of the Weyl tensor. In particular, we construct unimodular Lie groups
whose Weyl or Cotton-York tensors have the symmetries of conformally
transversally anisotropic manifolds, but which do not admit limiting Carleman
weights.
