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.