  For a class of multiparameter statistical models based on $N^2\times N^2$
braid matrices the eigenvalues of the transfer matrix ${\bf T}^{(r)}$ are
obtained explicitly for all $(r,N)$. Our formalism yields them as solutions of
sets of linear equations with simple constant coefficients. The role of
zero-sum multiplets constituted in terms of roots of unity is pointed out and
their origin is traced to circular permutations of the indices in the tensor
products of basis states induced by our class of ${\bf T}^{(r)}$ matrices. The
role of free parameters, increasing as $N^2$ with $N$, is emphasized
throughout. Spin chain Hamiltonians are constructed and studied for all $N$.
Inverse Cayley transforms of Yang-Baxter matrices corresponding to our braid
matrices are obtained for all $N$. They provide potentials for factorizable
$S$-matrices. Main results are summarized and perspectives are indicated in the
concluding remarks.
