Efficient factorization in some classes of second-order Holder matrix-functions

\selectlanguage{english} A homogeneous linear conjugate problem on a closed contour for a two-dimensional piecewise analytic vector is considered. Each solution of the problem is given a pair of functions, which are relations of limit values on the contour of the corresponding components of this solution. The relations connecting the elements of the $H$-continuous matrix-functions of the problem, providing the existence of its two solutions, for which the corresponding components of the pair differ by rational multipliers, and the problem itself admits a solution in closed form, are specified.