© 2018, Pleiades Publishing, Ltd. 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 Γ 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+∪ Γ and the determinant of the matrix of their components omitting those with index j is nonvanishing on Γ ∪ D−{∞}, where k, j= 1 , n¯ , then the canonical system of solutions to the linear conjugation problem can be constructed in closed form.