An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation $Lu=0$, where $L$ is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if $\Omega$ is an open subset of the plane with smooth boundary, $u\in C^1(\Omega)$ satisfies $Lu=0$ on $\Omega$, has tempered growth at the boundary, and its weak boundary value is a measure $\mu$, then $\mu$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partial\Omega$.