Geodesic
Systems of Linear Differential Equations. Integrable Combinations. Part I
Matematičeskoe obrazovanie, Tome 36 (2006) no. 1, pp. 2-9.

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For a linear system dy/dx = Ay the scalar product (a, y ) , where $\alpha$ is an eigenvector or a root vector of the conjugate operator $A^T$, satisfies a linear equation of the first order. This observation provides a method of integrating the original system. 
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V. V. Ivlev; E. Grjibovskaya. Systems of Linear Differential Equations. Integrable Combinations. Part I. Matematičeskoe obrazovanie, Tome 36 (2006) no. 1, pp. 2-9. http://geodesic.mathdoc.fr/item/MO_2006_36_1_a0/