Symmetry and perturbation of the vector nonlinear Schrِdinger equation

We show that the symmetry algebra of the N-component vector nonlinear Schrödinger (NLS) equation is given by the loop algebra of su(N)⊕u(1). The recursion operator and the Virasoro type algebra of the master symmetries (suitable for any N or even for an operator NLS equation) are also constructed. We have shown that a general SU(N) invariant (third-order) correction of the vector NLS equation can be moved in the next order of perturbation theory, i.e. the corresponding normal form of the equation is shown to be integrable.