  We analyse the famous Baxter's $T-Q$ equations for $XXX$ ($XXZ$) spin chain
and show that apart from its usual polynomial (trigonometric) solution, which
provides the solution of Bethe-Ansatz equations, there exists also the second
solution which should corresponds to Bethe-Ansatz beyond $N/2$. This second
solution of Baxter's equation plays essential role and together with the first
one gives rise to all fusion relations.
