Another comment on the Baxter condition for commutativity of transfer matrices

The Baxter condition has long been known to be sufficient for commutativity of transfer matrices. It was shown earlier that under an invariant subspace assumption the Baxter condition is also necessary. Although the invariant subspace condition holds genericly, it is very awkward to verify. Here we replace it by a symmetry condition. Namely we show that if the Boltzmann weights of two vertex models possess certain symmetry then the Baxter condition is necessary for commutativity of their transfer matrices.