A Frobenius integrability theorem for plane fields generated by quasiconformal deformations

We generalize the classical Frobenius integrability theorem to plane fields of class $C^Q$, a regularity class introduced by Reimann [Rei76] for vector fields in Euclidean spaces. A $C^Q$ vector field is uniquely integrable and its flow is a quasiconformal deformation. We show that an a.e. involutive $C^Q$ plane field (defined in a suitable way) in $\mathbb{R}^n$ is integrable, with integral manifolds of class $C^1$.