  We investigate certain classes of integrable classical or quantum spin
systems. The first class is characterized by the recursively defined property
$P$ saying that the spin system consists of a single spin or can be decomposed
into two uniformly coupled or disjoint subsystems with property $P$. For these
systems the time evolution can be explicitely calculated. The second class
consists of spin systems where all non-zero coupling constants have the same
strength (spin graphs) possessing $N-1$ independent, commuting constants of
motion of Heisenberg type. These systems are shown to have the above property
$P$ and can be characterized as spin graphs not containing chains of length
four. We completely enumerate and characterize all spin graphs up to N=5 spins.
Applications to the construction of symplectic numerical integrators for
non-integrable spin systems are briefly discussed.
