  We prove some part of the conjecture that regular Bethe ansatz eigenvectors
of the XXZ spin chain at roots of unity are highest weight vectors of the
$sl_2$ loop algebra. Here $q$ is related to the XXZ anisotropic coupling
$\Delta$ by $\Delta=(q+q^{-1})/2$, and it is given by a root of unity,
$q^{2N}=1$, for a positive integer $N$. We show that regular XXZ Bethe states
are annihilated by the generators ${\bar x}_k^{+}$'s, for any $N$. We discuss,
for some particular cases of N=2, that regular XXZ Bethe states are
eigenvectors of the generators of the Cartan subalgebra, ${\bar h}_k$'s. Here
the loop algebra $U(L(sl_2))$ is generated by ${\bar x}_k^{\pm}$ and ${\bar
h}_k$ for $k \in {\bf Z}$, which are the classical analogues of the Drinfeld
generators of the quantum loop algebra $U_q(L(sl_2))$. A representation of
$U(L(sl_2))$ is called highest weight if it is generated by a vector $\Omega$
which is annihilated by the generators ${\bar x}_k^{+}$'s and such that
$\Omega$ is an eigenvector of the ${\bar h}_k$'s. We also discuss the classical
analogue of the Drinfeld polynomial which characterizes the irreducible
finite-dimensional highest weight representation of $U(L(sl_2))$.
