  We study the implications of the regularization for the singular solutions on
the even(odd) length spin-1/2 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
\{\lambda_\alpha\} are invariant under the sign changes of their rapidities
{\lambda_\alpha\}=\{-\lambda_\alpha\} , then the Bethe ansatz equations are
reduced to a system of (M-2)/2 ((M-3)/2) 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 N= 3\left(2k+1\right) with k=1, 2, 3, \cdots. It is
also shown that there exist no singular solutions in the four down-spin sector
for some odd-length spin-1/2 XXX chains.
