The complex Frobenius theorem for rough involutive structures

We establish a version of the complex Frobenius theorem in the context of a complex subbundle $\mathcal{S}$ of the complexified tangent bundle of a manifold having minimal regularity. If the subbundle $\mathcal{S}$ defines the structure of a Levi-flat CR-manifold, it suffices that $\mathcal{S}$ be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on recent work of the authors.