Spectral problem for a two-component nonlinear Schr\"odinger equation in $2+1$ dimensions: Singular manifold method and Lie point symmetries

An integrable two-component nonlinear Schr"odinger equation in $2+1$ dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional Lie algebra. The main non-trivial similarity reductions associated to these symmetries are identified. The spectral parameter of the reduced spectral problem appears as a consequence of one of the symmetries.