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References

  1. V. I. Arnold and A. B. Givental. Symplectic Geometry, in the Book Dynamical Systems, vol. 4. Springer-Verlag, New York, 1988.

    Google Scholar 

  2. F. Magri: A geometrical approach to the nonlinear solvable equations. In: Boiti, M., Pempinelli, F., Soliani, G. (eds.) Nonlinear Evolution Equations and Dynamical Systems: Proceedings of the Meeting Held at the University of Lecce June 20–23, 1979. Lect. Notes Phys., 120, 233–263 (1980)

    Google Scholar 

  3. F. Magri. A simple model of the integrable Hamiltonian equation. J. Math. Phys., 19:1156, 1978.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. M. A. Semenov-Tian-Shansky. What is a classical r-matrix? Funct. Anal. Appl., 17(4):259–272, 1983.

    Article  Google Scholar 

  5. V. V. Trofimov and A. T. Fomenko. Algebra and Geometry of the Integrable Hamiltonian Differential Equations. Factorial, Minsk, 1995.

    Google Scholar 

  6. R. Abraham and J. E. Marsden. Foundations of mechanics, Advanced Book Program. Benjamin/Cummings Publishing, Menlo Park, CA, 1978.

    Google Scholar 

  7. J. F. Pommaret. Systems of Partial Differential Equations and Lie Pseudogroups. Mathematics and its applications. Gordon and Breach Science Publishers, New York, London, Paris, 1978.

    Google Scholar 

  8. I. V. Krasil’shchik, V. V. Lichagin, and A. M. Vinogradov. Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York, 1986.

    MATH  Google Scholar 

  9. P. J. Olver. Applications of Lie Groups to Differential Equations. Springer, Berlin, 2000.

    MATH  Google Scholar 

  10. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov. Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties. Russ. Math. Sur., 31(1):59–146, 1976. English translation from: Uspekhi Mat. Nauk, 62:6 (1977), 183–208.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Adler, P. Vanhaecke, and P. Van Moerbeke. Algebraic Integrability, Painlevé Geometry and Lie Algebras. Springer, Berlin-Heidelberg-New York, 2004.

    Google Scholar 

  12. M. Blaszak. Multi-Hamiltonian Theory of Dynamical Systems. Springer-Verlag, Berlin, Heidelberg, New York, 1998.

    MATH  Google Scholar 

  13. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math., 53: 249–315, 1974.

    MathSciNet  Google Scholar 

  14. V. S. Gerdjikov and P. P. Kulish. The Generating operator for n × n linear system. Physica D: Nonl. Phen., 3D(3):549–564, 1981.

    Article  ADS  MathSciNet  Google Scholar 

  15. V. S. Gerdjikov and A. B. Yanovski. Gauge covariant formulation of the generating operator. 1. The Zakharov–Shabat system. Phys. Lett. A, 103(5):232–236, 1984.

    Article  ADS  MathSciNet  Google Scholar 

  16. V. S. Gerdjikov and A. B. Yanovski. Gauge covariant formulation of the generating operator. 2. Systems on homogeneous spaces. Phys. Lett. A, 110(2):53–58, 1985.

    Article  ADS  MathSciNet  Google Scholar 

  17. V. S. Gerdjikov and A. B. Yanovski. Gauge covariant theory of the generating operator. I. Commun. Math. Phys., 103(4):549–568, 1986.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. V. E. Zakharov and L. A. Takhtadjan. Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys., 38(1):17–23, 1979.

    Article  Google Scholar 

  19. A. B. Yanovski. Gauge-covariant approach to the theory of the generating operators for soliton equations. PhD thesis, Autoreferat of a PhD Thesis, Joint Institute for Nuclear Research (JINR) 5-87-222, Dubna, 1987.

    Google Scholar 

  20. A. B. Yanovski. Generating operators for the generalized Zakharov–Shabat system in canonical and pole gauge. The sl(3C) case. Universität Leipzig, Naturwissenchaftlich Theoretisches Zentrum Report no. 20, 1993.

    Google Scholar 

  21. A. B. Shabat. An inverse scattering problem. Diff. Equ., 15(10):1299–1307, 1979.

    MathSciNet  Google Scholar 

  22. V. S. Shchesnovich and J. Yang. General soliton matrices in the Riemann-Hilbert problem for integrable nonlinear equations. ArXiv:nlin. SI/0306027, 2003.

    Google Scholar 

  23. A. O. Barut and R. RâFczka. Theory of Group Representations and Applications. World Scientific, Singapore, 1986.

    MATH  Google Scholar 

  24. M. Goto and F. Grosshans Semisimple Lie algebras, volume 38 of Lecture Notes in Pure and Applied Mathematics. M. Dekker Inc., New York and Basel, 1978.

    Google Scholar 

  25. R. Beals and R. R. Coifman. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math., 37:39–90, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Beals and R. R. Coifman. Inverse scattering and evolution equations. Commun. Pure Appl. Math., 38(1):29–42, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Beals and R. R. Coifman. The D-bar approach to inverse scattering and nonlinear evolutions. Physica D, 18(1–3):242–249, 1986.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. R. Beals and R. R. Coifman. Scattering and inverse scattering for first-order systems: II. Inverse Probl., 3(4):577–593, 1987.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. R. Beals and R. R. Coifman. Linear spectral problems, non-linear equations and the ∥-method. Inverse probl., 5(2):87–130, 1989.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. V. S. Gerdjikov and A. B. Yanovski. Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys., 35:3687–3721, 1994.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. G. Landi, G. Marmo, and G. Vilasi. Remarks on the complete integrability of dynamical systems with fermionic variables. J. Phys. A: Math. Gen., 25(16):4413–4423, 1992.

    Google Scholar 

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Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). Introduction. In: Gerdjikov, V., Vilasi, G., Yanovski, A. (eds) Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77054-1_10

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