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In this paper we prove the existence of an exponentially localized stationarysolution for a two-dimensional cubic Dirac equation. It appears as an effectiveequation in the description of nonlinear waves for some Condensed Matter(Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. Thenonlinearity is of Kerr-type, that is of the form |$\psi$| 2 $\psi$ and thusnot Lorenz-invariant. We solve compactness issues related to the criticalSobolev embedding H 1 2 (R 2 , C 2) $\rightarrow$ L 4 (R 2 , C 4) thanks to aparticular radial ansatz. Our proof is then based on elementary dynamicalsystems arguments. Contents

Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity