  We examine a simple heuristic test of integrability for quantum chains. This
test is applied to a variety of systems, including a generic isotropic spin-1
model with nearest-neighbor interactions and a multiparameter family of
spin-1/2 models generalizing the XYZ chain, with next-to-nearest neighbor
interactions and bond alternation. Within the latter family we determine all
the integrable models with an o(2) symmetry.
