  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.
