  We formulate an equivalence between the 2-dim $\sigma$-model spectrum
expanded on a non-trivial massive vacuum and a classical particle Hamiltonian
with variable mass and potential. By considering methods of analytic Galoisian
non-integrability on appropriate geodesics of the Hamiltonian system we
algebraically constrain the particle masses at fixed time, such that
integrability is allowed. Through our equivalence this explicitly constrains
the masses of the excited spectrum of the dual 2-dim theory in such a way to
imply the S-matrix factorization and no particle production. In particular, the
integrability of the classical particle system, implies the factorization of
the S-matrix in the dual quantum 2-dim theory. Our proposal provides also
non-trivial evidence without any assumptions, on the connection between
integrability and S-matrix factorization for large class of theories with
interactions that break Lorentz invariance.
