Some classes of linear conjugation problems for a two-dimensional vector resolvable in a closed form

We consider the structure of the set of piecewise-meromorphic solutions to the linear conjugation problem for a two-dimensional vector and study its connection with the fractional linear conjugation problem. We prove that if there exists a piecewise-meromorphic solution to any of these problems then one can construct a canonical system of solutions to the linear conjugation problem and define classes of problems solvable in a closed form.