  We establish a solvable Heisenberg-Ising model on a spin-1 Ni-containing
polymer chain, $[Ni (NN'-dmen) (\mu-N_3)_2]$, with $NN'-dmen$ being
$NN'-dimethylethylenediamine$, that fully covers the interaction
characteristics of the material and by which, we can characterize all the
peculiar magnetic features of the polymer, which has been partly studied in
experiment. By purely analytical calculations, we can see that the
magnetization exhibits three plateaus at zero, mid, and 3/4 of the saturation
value at low temperatures below 2 K. The corresponding featuring peaks of
magnetic susceptibility are clearly shown. The model also displays plateaus in
thermal entanglement that captures the one-to-one correspondence between
thermal entanglement plateaus and those of the magnetization. The calculations
are done by the transfer matrix technique.
