Jordan blocks and the Bethe ansatz II: The eclectic spin chain beyond $K=1$

We continue the classification of the Jordan chains of the eclectic three state spin chain that we started in our previous article. Following the same steps, we construct the generalised eigenvectors of this spin chain by computing the strongly twisted limit of linear combinations of eigenvectors of a twisted XXX $SU(3)$ spin chain. We show that this classification problem can be mapped to the computation of the number of positive integer solutions of a system of linear equations.