Investigation of the exact integrability of the multiwave Schrödinger equation

It is shown that the multiwave nonlinear Schrödinger equation, which describes evolution of several quasi-monochromatic waves with equal group velocities, is not exactly integrable: it does not have infinite sequence of local conservation laws and symmetries. Exact integrability of the system of equations $w_t^i=\alpha_iw_{xx}^i+a_{klm}^iw^kw^lw^m$ with the nondegenerate diagonal matrix at the second-order derivative is studied.