On the solvability of one system of linear differential equations in the space of functions limited for the entire axis

The issue of unique solvability in the space of functions limited for the entire axis for one system of linear differential equations with unlimited coefficients is considered. When setting tasks, the matrix of coefficients is divided into two matrices, the matrix of "senior" and the matrix of "lower" coefficients. It is assumed that the spectrum of the matrix of "senior" coefficients has an intersection with the imaginary axis. The conditions for the matrix of "lower" coefficients are revealed, the fulfillment of which ensures the unique solvability of the system in the space of functions limited for the entire axis. Revealed conditions are written out using the relationship between the "senior and junior" coefficients of the system.