We establish a ``low rank property'' for Sobolev mappings that almost everywhere solve a special nonlinear system of PDEs. This system, associated to a nonintegrable tangent distribution, implies the so-called contact property of its solutions. The proof of this property relies on a "special weakly exterior differentiation'' performed through a blow-up procedure. As an application, we give a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. These results are a joint work with J. Malý and S. Mongodi.
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