Contribution to the general linear conjugation problem for a piecewise analytic vector

Establishing an analogy between the theories of Riemann–Hilbert vector problem and linear ODEs, for the $n$-dimensional homogeneous linear conjugation problem on a simple smooth closed contour $\Gamma$ partitioning the complex plane into two domains $D^+$ and $D^-$ we show that if we know $n-1$ particular solutions such that the determinant of the size $n-1$ matrix of their components omitting those with index $k$ is nonvanishing on $D^+\cup\Gamma$ and the determinant of the matrix of their components omitting those with index $j$ is nonvanishing on $\Gamma\cup D^-\setminus\{\infty\}$, where $k,j=\overline{1,n}$, then the canonical system of solutions to the linear conjugation problem can be constructed in closed form.