  We establish a version of the complex Frobenius theorem in the context of a
complex subbundle S of the complexified tangent bundle of a manifold, having
minimal regularity. If the subbundle S defines the structure of a Levi-flat
CR-manifold, it suffices that 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 previous recent work of the authors.

Differential Geometry;Analysis of PDEs;Complex Variables