A method of defining a class of linear conjugation problems for two-dimensional vector with closed-form solutions

The problem of linear conjugation for a two-dimensional piecewise analytic vector was reduced to an equivalent problem of fractional linear conjugation, and a connection between their solutions was established. It was shown that, once a particular solution of either problem is known, the canonical system of solutions of the linear conjugation problem can be written in closed form. The relations were specified between the elements of the H¨older matrix-function of the linear conjugation problem under which the fractional linear conjugation problem has a rational solution, thus enabling a closed solution of the linear conjugation problem.