On $\mathbb{R}$-linear conjugation problem on the unit circle

A new method of finding a solution to the $\mathbb{R}$-linear conjugation problem on the unit circle is proposed. The problem is studied under the assumption that its main coefficient is a segment of the Fourier series. The applied method is based on reducing the considered problem to the vector-matrix boundary value problem and applying the recently suggested generalization of G. N. Chebotarev's approach to the factorization of triangular matrix functions to its matrix coefficient.