The variety of solutions of the singular generalized Cauchy-Riemann system (Q1358472)

We study the homogeneous equation \(2\overline z\partial_{\overline z}w- B(z)\overline w=0\), \(z\in G\), in which \(w(z)\) is a complex-valued function and \(B(z)\) is a given function of the variable \(z=x+iy= re^{i\varphi}\), and the singular point \(z=0\) lies in the interior of the domain \(G\). We assume that \(B(z)=b(\varphi)+ B_0(z)\), where \(b(\varphi)\) can be represented by a truncated Fourier series \[ b(\varphi)= \sum^m_{k=-m_0} b_ke^{ik\varphi}, \] \(m_0\) and \(m\) are nonnegative integers, \(B_0(z)\in C^\infty(G)\), and \(B_0(z)= O(|z|^\alpha)\), \(\alpha>0\), \(z\to 0\). A generic condition on \(b(\varphi)\) is stated which implies that there do not exist nontrivial solutions.