On the construction of regular solutions for a class of generalized Cauchy–Riemann systems with coefficients bounded on the entire plane

This article explores the generalized Cauchy–Riemann system on the entire complex plane. The coefficient for the conjugation of the desired function belongs to the Hölder space and, for $|z|>1$, equals $e^{im\varphi}$, where $m$ is an integer. For $m\le 0$, the system was shown to have no nonzero solutions that grow no faster than a polynomial. For $m\ge 0$, the complete set of regular solutions, i.e., those without singularities in the finite part of the plane, was constructed. The obtained solutions were expressed as series of Bessel functions of an imaginary argument. From the resulting set, the solutions bounded on the entire plane were distinguished, and the dimension of the real linear space of these solutions, which equals $m$, was determined.