The variety of solutions of the singular generalized Cauchy–Riemann System

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 $C^\infty(G)$.