= Solution
For definiteness take $f$ in the <Schwartz space>; sufficiently smooth compactly supported data are also adequate. These hypotheses make the line integrals, <Cauchy principal values> and decay estimates below well defined. Write $\lambda=re^{i\theta}$ with $r>0$, $r\ne1$, and introduce orthonormal local coordinates
$$
\tau=x_1\cos\theta+x_2\sin\theta,\qquad
\rho=-x_1\sin\theta+x_2\cos\theta.
$$
Let $F(\tau,\rho,\theta)=f(x_1,x_2)$, $a_r=(r+r^{-1})/2$ and $b_r=(r-r^{-1})/2$. Substitution into the directional <transport equation> gives
$$
L_\lambda=a_r\partial_\tau-i b_r\partial_\rho.
$$
Now make the invertible real-linear change of coordinates
$$
\boxed{z=\tau-i\frac{a_r}{b_r}\rho,\qquad \nu(r)=2a_r=r+r^{-1}.}
$$
Since $\operatorname{Im}z=-a_r\rho/b_r$, its <Wirtinger derivative> is $\partial_{\bar z}=(\partial_\tau-i b_r\partial_\rho/a_r)/2$. Hence $L_\lambda=\nu(r)\partial_{\bar z}$, as required. The change becomes singular at $r=1$, where the complex operator degenerates to a real directional derivative. Also the original equation contains $\lambda^{-1}$, so $\lambda=0$ is excluded from the differential equation; the normalized solution will extend analytically there.
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