We consider two one-parameter families of two-dimensional relativistic
factorized scattering theories, which are deformations of the ones ch
aracteristic for the two-dimensional O(3) sigma models with theta = 0
and theta = pi. The Bethe ansatz technique is applied to these two fam
ilies to justify their interpretation as the factorized scattering the
ories of certain O(2)-symmetric deformations of the O(3) sigma model (
the sausage model). The result suggests that the sausage model is inte
grable at the topological angle values theta = 0 or theta = pi.