Factorization of the scattering matrix and the location of the eigenvalues of the Manakov-Zakharov-Shabat system

We show the scattering matrix associated to the Manakov-Zakharov-Shabat (MZS) system can be factorized as the product of two scattering matrices associated to the Zakharov-Shabat (ZS) system of the Nonlinear Schrödinger (NLS) equation, whenever the initial conditions of the Manakov system have disjoint support. Moreover, if these initial conditions are assumed to be single-lobe, the eigenvalues of the MZS system are purely imaginary.