  In this paper we study the attenuated $X$-ray transform of 2-tensors
supported in strictly convex bounded subsets in the Euclidean plane. We
characterize its range and reconstruct all possible 2-tensors yielding
identical $X$-ray data. The characterization is in terms of a Hilbert-transform
associated with $A$-analytic maps in the sense of Bukhgeim.
