n-1 independent first integrals for linear differential systems in Rn and Cn


We prove that every linear system with constant coefficients on Rn or Cn is Darboux integrable by providing a complete explicit list ofn−1 independent Darboux first integrals.

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  1. 1.

    R. M. Bulatovich,On the continuability of a gyroscopic integral from a linear system to a nonlinear system, Teor. Primen. Meh.21 (1995), 1–7.

    MathSciNet  Google Scholar 

  2. 2.

    J. Chavarriga, J. Llibre andJ. Sotomayor,Algebraic solutions for polynomial systems with emphasis in the quadratic case, Exp. Math.15 (1997), 161–173.

    MATH  MathSciNet  Google Scholar 

  3. 3.

    C. Christopher andJ. Llibre,Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. of Differential Equations16 (2000), 5–19.

    MATH  MathSciNet  Google Scholar 

  4. 4.

    C. Christopher, J. Llibre and J. V. Pereira,Multiplicity of invariant algebraic curves and Darboux integrability, Preprint 2000.

  5. 5.

    G. Darboux,Memoire sur les équations differentielles algébrique du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série2 (1878), 60–96, 123–144, 151–200.

    Google Scholar 

  6. 6.

    H. Flaschka, A.C. Newell andM. Tabor,Integrability, In What is Integrability? Ed. V.E. Zakharov, Springer-Verlag, Series in Nonlinear Dynamics, (1991), 71–114.

    Google Scholar 

  7. 7.

    J. P. Jouanolou,Equations de Pfaff Algébriques, Lect. Notes in Math.708, Springer-Verlag, Berlin, (1979).

    Google Scholar 

  8. 8.

    S. Maeda,On inheritance of quadratic first integral of linear system via Runge-Kutta methods, J. Math. Tokushima Univ.31 (1997), 63–67.

    MATH  MathSciNet  Google Scholar 

  9. 9.

    J. Moulin Ollagnier andJ.M. Strelcyn,On first integrals of Linear systems, Frobenius integrability theorem and linear representations of Lie algebras, Lect. Notes in Math.1455, Springer-Verlag (1991), 243–271.

    Article  MathSciNet  Google Scholar 

  10. 10.

    A. Nowicki,On the nonexistence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107–120.

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    M. J. Prelle andM. F. Singer,Elementary first integrals of differential equations, Trans. Amer. Math. Soc.279 (1983), 215–229.

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    W. T. Van Horssen,On integrating factors for ordinary differential equations, Nieuw Arch. Wisk.15 (1997), 15–26.

    MATH  MathSciNet  Google Scholar 

  13. 13.

    J. A. Weil,First integrals and Darboux polynomials of homogenous linear differential systems, Lect. Notes in Computer Science948, Springer-Verlag (1995), 469–484.

    MathSciNet  Google Scholar 

  14. 14.

    V.E. Zakharov,What is integrability?, Springer Series in Nonlinear Dynamics, Springer-Verlag, (1991).

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Correspondence to Manuel Falconi.

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Dedicated to Jorge Sotomayor on his 60th birthday

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Falconi, M., Llibre, J. n-1 independent first integrals for linear differential systems in Rn and Cn . Qual. Th. Dyn. Syst 4, 233–254 (2004). https://doi.org/10.1007/BF02970860

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Key Words

  • First integral
  • Darboux integrability
  • invariant surface