  Our starting point is a class of braid matrices, presented in a previous
paper, constructed on a basis of a nested sequence of projectors. Statistical
models associated to such $N^2\times N^2$ matrices for odd $N$ are studied
here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature
of our models, setting them apart from other well-known ones. There are $N$
possible states at each site. The trace of the transfer matrix is shown to
depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute
the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of
parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or
lower dimensional multiplets corresponding to factors of the order $r$ when $r$
is not a prime number. The modulus of any eigenvalue is of the form
$e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters,
$\theta$ being the spectral parameter. For $r$ a prime number an amusing
relation of the number of multiplets with a theorem of Fermat is pointed out.
Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices
are constructed starting from our braid matrices. Perspectives are discussed.
