  We explicitly calculate the $AdS_2 \times S^2 \times T^6$ transfer-matrix
eigenvalues in the massless sector using the exact integrable S-matrix, for up
to 5 particles. This enables us to conjecture the general pattern. We use the
conjectured form of the eigenvalues to write down a set of massless Bethe
ansatz equations. The same procedure applies to the relativistic as well as to
the non-relativistic situation. In the relativistic case, the right and left
modes decouple. We speculate that the relativistic massless Bethe ansatz we
obtain in that case might capture the integrable structure of an underlying 2D
critical theory. We finally take advantage of some remarkable simplifications
to make progress in the massive case as well.
