Geodesic
The variety of solutions of the singular generalized Cauchy--Riemann System
Matematičeskie zametki, Tome 59 (1996) no. 2, pp. 278-283

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We prove that the equation $$ 2\overline z\partial_{\overline z}w-\bigl(b(\varphi)+B(z)\bigr)\overline w=0,\quad z\in G, $$ in which $B(z)\in C^\infty(G)$, $B_0(z)=O(|z|)^\alpha)$, $\alpha>0$, $z\to0$, and $$ b(\varphi)=\sum_{k=-m_0}^mb_ke^{ik\varphi}, $$ does not have nontrivial solutions in the class $C^\infty(G)$. 
@article{MZM_1996_59_2_a12,
     author = {Z. D. Usmanov},
     title = {The variety of solutions of the singular generalized {Cauchy--Riemann} {System}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {278--283},
     publisher = {mathdoc},
     volume = {59},
     number = {2},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_2_a12/}
}
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Z. D. Usmanov. The variety of solutions of the singular generalized Cauchy--Riemann System. Matematičeskie zametki, Tome 59 (1996) no. 2, pp. 278-283. http://geodesic.mathdoc.fr/item/MZM_1996_59_2_a12/