 Rank-one level curves from a characteristic differential equation

For a smooth map $F=(f,g)$ on the plane with zero Jacobian determinant and $f_y\ne 0$, set $e=g_y/f_y$. The determinant identity gives $\nabla g=e\nabla f$. Along a curve solving $y^{\prime }=-f_x/f_y$, both $f$ and $g$ have zero derivative. Local existence and uniqueness for this smooth differential equation through every nearby point produce a local foliation by curves of constant $F$.