  A model for planar phenomena introduced by Jackiw and Pi and described by a Lagrangian including a Chern-Simons term is considered. The associated equations of motion, among which a 2+1 gauged nonlinear Schr\"odinger equation, are rewritten into a gauge independent form involving the modulus of the matter field. Application of a Painlev\'e analysis, as adapted to partial differential equations by Weiss, Tabor and Carnevale, shows up resonance values that are all integer. However, compatibility conditions need be considered which cannot be satisfied consistently in general. Such a result suggests that the examined equations are not integrable, but provides tools for the investigation of the integrability of different reductions. This in particular puts forward the familiar integrable Liouville and 1+1 nonlinear Schr\"odinger equations. 
Painlev\'e analysis and integrability properties of a $2+1$   nonrelativistic field theory.