Markushevich's ill-posed boundary-value problem for multiply connected domains with circular boundaries

We consider a little-known boundary-value problem of linear conjugation in the theory of functions of a complex variable in multiply connected domains. Although it was posed in a general form by Markushevich in 1946 and used in Vekua's geometric studies, it is still poorly understood. The problem is ill-posed: it loses solubility after arbitrarily small perturbations of the coefficient. Under certain simple conditions, we solve the problem completely by establishing necessary and sufficient criteria for its solubility and giving a construction for its solutions.