The authors consider a \(2+1\)-dimensional field theory described by a nonlinear Schrödinger equation, with ordinary derivatives replaced by gauge covariant derivatives, supplemented by (Chern-Simons) constraints on the gauge and matter fields. Their concern is to establish the integrability of this theory. They proceed by applying the J. Weiss, M. Tabor and G. Carnevale [J. Math. Phys. 24, 522–526 (1983; Zbl 0514.35083)] generalization of the Painlevé test for partial differential equations. They effect this on a gauge independent form of the equations to obtain integer resonances of orders \(1, 2, 3\) and 4. However compatibility conditions on the arbitrary functions introduced in the analysis are not satisfied and (even conditionally) the test is not passed. The authors close by pointing out that there exist several reductions which may, however, be integrable.