  We study the integrability of the general two-dimensional Zakharov-Shabat
systems, which appear in application of the inverse scattering transform (IST)
to an important class of nonlinear partial differential equations (PDEs) called
integrable systems, in the meaning of differential Galois theory, i.e., their
solvability by quadrature. It becomes a key for obtaining analytical solutions
to the PDEs by using the IST. For a wide class of potentials, we prove that
they are integrable in that meaning if and only if the potentials are
reflectionless. It is well known that for such potentials particular solutions
called n-solitons in the original PDEs are yielded by the IST.
