On an analogue of the Riemann–Hilbert problem for a non-linear perturbation of the Cauchy–Riemann operator

We consider a non-linear perturbation of a famous Riemann–Hilbert problem on the recovering of a holomorphic function in a domain via its real part on the boundary. We get an information on the local structure of the solutions and give sufficient conditions for their real analyticity. A simple instructive example is considered.