  In this paper we prove the existence of an exponentially localized stationary
solution for a two-dimensional cubic Dirac equation. It appears as an effective
equation in the description of nonlinear waves for some Condensed Matter
(Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. The
nonlinearity is of Kerr-type, that is of the form |$\psi$| 2 $\psi$ and thus
not Lorenz-invariant. We solve compactness issues related to the critical
Sobolev embedding H 1 2 (R 2 , C 2) $\rightarrow$ L 4 (R 2 , C 4) thanks to a
particular radial ansatz. Our proof is then based on elementary dynamical
systems arguments. Contents
