  The universal formulation of spin exchange models related to Calogero-Moser
models implies the existence of integrable hierarchies, which have not been
explored. We show the general structures and features of the spin exchange
model hierarchies by taking as examples the well-known Heisenberg spin chain
with the nearest neighbour interactions. The energy spectra of the second
member of the hierarchy belonging to the models based on the $A_r$ root systems
$(r=3,4,5)$ are explicitly and {\em exactly} evaluated. They show many many
interesting features and in particular, much higher degree of degeneracy than
the original Heisenberg model, as expected from the integrability.
