Linear autonomy conditions for the basic Lie algebra of a system of linear differential equations

Linear autonomy conditions for all operators of the basic Lie algebra of a system of first-order linear differential equations with constant coefficients over the field of complex numbers are suggested. The system with constant complex matrices is considered such that the matrices are linearly dependent. For a certain variable equal to 1, the basic Lie algebra of system is x-autonomous if and only if the matrix of this system does not satisfy the quadratic equation. If the system is not equivalent to an exceptional system and admits a non-x-autonomous operator, then there exists a matrix of rank 1 that commutes with every matrix of this system. All operators admitted by the system are linearly autonomous if the system is equivalent to an exceptional systems.