Abstract
We construct Lax pairs for linear Hamiltonian systems of differential equations. In particular, the Gröbner bases are used for computations. It is proved that the mappings in the construction of Lax pairs are Poisson. Under study are the various properties of first integrals of the system which are obtained from Lax pairs.
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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 760–776.
The first author was supported by the Russian Foundation for Basic Research (Grants 16-01-00378a and 16-51-55012 China-a). The second author was partially supported by the Laboratory of Mirror Symmetry NRU HSE (RF Government Grant, Ag. No. 14.641.31.0001).
We are grateful to A. N. Parshin for posing the problem and numerous discussions. The starting point of our research was the report by V. V. Kozlov “The Symplectic Geometry of Linear Hamiltonian Systems and the Solution of Algebraic Equations” at the joint seminar of the Departments of Algebra and Algebraic Geometry at the Steklov Mathematical Institute in September 2017.
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Zheglov, A.B., Osipov, D.V. Lax Pairs for Linear Hamiltonian Systems. Sib Math J 60, 592–604 (2019). https://doi.org/10.1134/S0037446619040050
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DOI: https://doi.org/10.1134/S0037446619040050