  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.
