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We consider a total reduction of a nonhomogeneous linear system of operator  equations with the system matrix in the companion form. Totally reduced  system obtained in this manner is completely decoupled, i.e., it is a system  with separated variables. We introduce a method for the total reduction, not  by a change of basis, but by finding the adjugate matrix of the  characteristic matrix of the system matrix. We also indicate how this  technique may be used to connect differential transcendence of the solution  with the coefficients of the system.

Total reduction of linear systems of operator equations with the system matrix in the companion form