  In this paper we investigate the quantum dynamics of two spin-1 systems,
$\vec{\textbf{S}}_1$ and $\vec{\textbf{S}}_2$, adopting a generalized
$(\vec{\textbf{S}}_1+\vec{\textbf{S}}_2)^2$-nonconserving Heisenberg model. We
show that, due to its symmetry property, the nine-dimensional dynamics of the
two qutrits exactly decouples into the direct sum of two sub-dynamics living in
two orthogonal four- and five-dimensional subspaces. Such a reduction is
further strengthened by our central result consisting in the fact that in the
four-dimensional dynamically invariant subspace, the two qutrits quantum
dynamics, with no approximations, is equivalent to that of two non interacting
spin 1/2's. The interpretative advantages stemming from such a remarkable and
non-intuitive nesting are systematically exploited and various intriguing
features consequently emerging in the dynamics of the two qutrits are deeply
scrutinised. The possibility of exploiting the dynamical reduction brought to
light in this paper for exactly treating as well time-dependent versions of our
Hamiltonian model is briefly discussed.
