  We conjecture that the free-fermion part of the eigenspectrum observed
recently for the $SU_q(N)$ Perk-Schultz spin chain Hamiltonian in a finite
lattice with $q=\exp (i\pi (N-1)/N)$ is a consequence of the existence of a
special simple eigenvalue for the transfer matrix of the auxiliary
inhomogeneous $SU_q(N-1)$ vertex model which appears in the nested Bethe ansatz
approach. We prove that this conjecture is valid for the case of the SU(3) spin
chain with periodic boundary condition. In this case we obtain a formula for
the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model
($q=\exp (2 i \pi/3)$), which permit us to find one by one all components of
this eigenvector and consequently to find the eigenvectors of the free-fermion
part of the eigenspectrum of the SU(3) spin chain. Similarly as in the known
case of the $SU_q(2)$ case at $q=\exp(i2\pi/3)$ our numerical and analytical
studies induce some conjectures for special rates of correlation functions.
