Analytic nonintegrability and S -matrix factorization

We formulate an equivalence between the 2-dim $\ensuremath{\sigma}$-model spectrum expanded on a nontrivial massive vacuum and classical particle Hamiltonian with variable mass potential. By considering methods of analytic Galoisian nonintegrability appropriate geodesics system we algebraically constrain masses at fixed time, such that integrability is allowed. Through our equivalence, this explicitly constrains excited dual theory in way to imply $S$-matrix factorization no production. In particular, implies quantum theory. Our proposal provides also evidence connection for large class theories interactions break Lorentz invariance.