On the X-ray transform of planar symmetric 2-tensors

In this paper we study the attenuated X-ray transform of 2-tensors supported in convex bounded subsets with sufficiently smooth boundary 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.