Some classes of linear conjugation problems for a three-dimensional vector that are solvable in closed form

We consider the structure of the set of piecewise meromorphic solutions for a homogeneous linear conjugation problem for a three-dimensional vector. We show that, in the presence of two piecewise meromorphic solutions to the linear conjugation problem, it is possible to construct a canonical system of solutions to the linear conjugation problem and distinguish some classes of problems solvable in closed form.