  Most of the work done in the past on the integrability structure of the
Classical Heisenberg Spin Chain (CHSC) has been devoted to studying the $su(2)$
case, both at the continuous and at the discrete level. In this paper we
address the problem of constructing integrable generalized ''Spin Chains''
models, where the relevant field variable is represented by a $N\times N$
matrix whose eigenvalues are the $N^{th}$ roots of unity. To the best of our
knowledge, such an extension has never been systematically pursued. In this
paper, at first we obtain the continuous $N\times N$ generalization of the CHSC
through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit
some explicit, and hopefully interesting, examples for $3\times 3$ and $4\times
4$ matrices; then, we discuss the much more difficult discrete case, where a
few partial new results are derived and a conjecture is made for the general
case.
