  We study the Cauchy problem for the (2+1) integrable nonlinear Schr\"odinger
equation by the inverse scattering transform (IST) method. This Cauchy problem
with given initial data and boundary data at infinity is reduced by IST to the
Cauchy problem for the linear Schr\"odinger equation, in which the potential is
expressed in terms of boundary data. The results on direct and inverse
scattering problems for a two-dimensional Dirac system with special potentials
are used and refined. The Cauchy problem admits an explicit solution if the IST
of the solution is an integral operator of rank 1. We give one such solution.
