The integrability is underpinned by the existence of large symmetry algebras for the different models. For the XXX case this is the Yangian , while in the XXZ case this is the quantum group , the q-deformation of the affine Lie algebra of , as explained in the notes by .
These appear through the transfer matrix, and the condition that the Bethe vectors are generated from a state  satisfying  corresponds to the solutions being part of a highest-weight representation of the extended symmetry algebras.