  We present a detailed analysis of the spin models with near-neighbors
interactions constructed in our previous paper [Phys. Lett. B 605 (2005) 214]
by a suitable generalization of the exchange operator formalism. We provide a
complete description of a certain flag of finite-dimensional spaces of spin
functions preserved by the Hamiltonian of each model. By explicitly
diagonalizing the Hamiltonian in the latter spaces, we compute several infinite
families of eigenfunctions of the above models in closed form in terms of
generalized Laguerre and Jacobi polynomials.
