Bethe equations `on the wrong side of the equator'

We analyse Baxter's famous T-Q equations for the XXX ( XXZ ) spin chain and show that, apart from its usual polynomial (trigonometric) solution, which provides the solution of Bethe ansatz equations, there also exists a second solution which should correspond to the Bethe ansatz beyond . This second solution of Baxter's equation plays an essential role and together with the first one gives rise to all fusion relations.