= Solution
Write $\chi=e^{-ikz-\lambda\bar z/(ik)}$. In terms of <Wirtinger derivatives>, the elliptic equation is $q_{z\bar z}=\lambda q$. If $W=A\,dz+B\,d\bar z$, its coefficients are $A=\chi q_z$ and $B=-\lambda\chi q/(ik)$. Direct differentiation gives
$$
A_{\bar z}=\chi\left(\lambda q-\frac{\lambda}{ik}q_z\right),\qquad B_z=\chi\left(\lambda q-\frac{\lambda}{ik}q_z\right).
$$
Therefore $dW=(B_z-A_{\bar z})dz\wedge d\bar z=0$, proving that \b[$W$ is a closed differential one-form]. The parameter restriction $k\ne0$ makes its coefficients well defined.
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