Geodesic
Systems of Linear Differential Equations. Integrable Combinations. Part I
Matematičeskoe obrazovanie, no. 1 (2006), pp. 2-9 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {V. V. Ivlev and E. Grjibovskaya},
     title = {Systems of {Linear} {Differential} {Equations.} {Integrable} {Combinations.} {Part} {I}},
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V. V. Ivlev; E. Grjibovskaya. Systems of Linear Differential Equations. Integrable Combinations. Part I. Matematičeskoe obrazovanie, no. 1 (2006), pp. 2-9. http://geodesic.mathdoc.fr/item/MO_2006_1_a0/