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We consider the structure of the set of piecewise meromorphic solutions to the linear conjugation problem for a two-dimensional vector and study its relations with the fractional-linear conjugation problem. We prove that under the assumption of the piecewise meromorphic solvability of any of these problems there exists a canonical system of solutions to the linear conjugation problem and describe classes of problems which are solvable in a closed form.
S. N. Kiyasov (Tue,) studied this question.