Abstract
We study the implications of the regularization for the singular solutions on the even(odd) length spin- XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions are invariant under the sign changes of their rapidities , then the Bethe ansatz equations are reduced to a system of equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form with . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin- XXX chains.
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