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References
F. Magri and C. Morosi. A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds. Quaderni del Dipartimento di Matematica, Università di Milano, 1984.
V. G. Drinfeld and V. V. Sokolov. Lie algebras and Korteweg-de Vries type equations. VINITI Series: Contemporary problems of mathematics. Recent developments. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985.
V. E. Zakharov and L. D. Faddeev. Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl., 5(4):280–287, 1971.
M. Adler. On a Trace Functional for Formal Pseudo-Differential Operators and the Symplectic Structure of the Korteweg–Devries Type Equation. Invent. Math., 50:219, 1978.
S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. American Mathematical Society, Providence, RI, 2001.
J. M. Souriau. Structure des systémes dynamiques. Dunod, Paris, 1970.
P. Libermann and C. M. Marle. Symplectic Geometry and Analytical Mechanics, volume 35 of Mathematics and its Applications. D. Reidel Publishing Co., Dordrecht, 1987.
J. E. Marsden. Lectures on Mechanics, volume 174 of London Mathematical Society, Lecture Note Series. Cambridge University Press, Cambridge, 1992.
J. E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 5(1):121–130, 1974.
J. P. Ortega and T. S. Ratiu. Momentum Maps and Hamiltonian Reduction, volume 222 of Progress in Mathematics. Birkhäuser, Boston, MA, 2004.
V. V. Trofimov and A. T. Fomenko. Algebra and Geometry of the Integrable Hamiltonian Differential Equations. Factorial, Minsk, 1995.
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.
Magri, F.: 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)
A. G. Reyman. Integrable Hamiltonian systems connected with graded Lie algebras. J. Sov. Math., 19:1507–1545, 1982.
A. G. Reyman. General Hamiltonian structure on polynomial linear problems and the structure of stationary equations. J. Sov. Math., 30(4):2319–2326, 1985.
A. G. Reiman and M. A. Semenov-Tyan-Shanskii. A family of Hamiltonian structures, hierarchy of Hamiltonians, and reduction for first-order matrix differential operators. Funct. Anal. Appl., 14(2):146–148, 1980.
A. G. Reiman and M. A. Semenov-Tyan-Shanskii. The jets algebra and nonlinear partial differential equations. Dokl. Akad. Nauk SSSR, 251(6):1310–1314, 1980.
L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.
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.
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.
V. S. Gerdjikov and A. B. Yanovski. Gauge covariant theory of the generating operator. I. Commun. Math. Phys., 103(4):549–568, 1986.
V. S. Gerdjikov and A. B. Yanovski. Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys., 35:3687–3721, 1994.
I. D. Iliev, E. Kh. Christov, and K. P. Kirchev. Spectral Methods in Soliton Equations, volume 73 of Pitman Monographs and Surveys in Pure and Applied Mathematics. John Wiley& Sons, New York, 1991.
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Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). Poisson–Nijenhuis structures Related to the Generalized Zakharov–Shabat System. 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_15
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