  Based on the Lieb-Schultz-Mattis construction we present a five parameter
family of Spin-1 Hamiltonians with degenerate groundstate. Starting from the
critical $SU(3)$ symmetric Hamiltonian, we look for those perturbations of the
$SU(3)$ symmetry, which leave the groundstate degenerate. We also discuss the
spin-3/2 $SU(4)$-case.
