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 $��$ is an open subset of the plane with smooth boundary, $u\in C^1(��)$ satisfies $Lu=0$ on $��$, has tempered growth at the boundary, and its weak boundary value is a measure $��$, then $��$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partial��$.

20 pages