Certain reductions of Hitchin systems of rank 2 and genera 2 and 3.

*(English. Russian original)*Zbl 1432.14030
Dokl. Math. 97, No. 2, 144-146 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 479, No. 3, 254-256 (2018).

In the present paper, the author considers reductions of Hitchin systems, i.e., their restrictions to invariant subvarieties of a positive codimension. He obtains some manageable rank 2, genera 2 and 3 systems, certain relations for their dynamical variables, and some particular solutions to the original systems. Certain reductions are shown to give an integrable system of 2, resp. 3, interacting points on the line. It is shown that the reduced systems are particular cases of a certain universal integrable system related to the Lagrange interpolation polynomial. Admissibility of the reduction is proved using computer technique. The corresponding codes are given in the text.

This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2, the author describes the reduction of the Hitchin systems of rank 2, genera 2 and 3, proves it to be an admissible reduction, observes that the reduced systems are completely integrable and finds some their particular solutions which are solutions to the original Hitchin system. The proof of admissibility is computational. The codes of the corresponding programs are given in Sections 4, 5. Section 3 is devoted to the integrable system related to the Lagrange interpolation polynomial.

This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2, the author describes the reduction of the Hitchin systems of rank 2, genera 2 and 3, proves it to be an admissible reduction, observes that the reduced systems are completely integrable and finds some their particular solutions which are solutions to the original Hitchin system. The proof of admissibility is computational. The codes of the corresponding programs are given in Sections 4, 5. Section 3 is devoted to the integrable system related to the Lagrange interpolation polynomial.

Reviewer: Ahmed Lesfari (El Jadida)

##### MSC:

14H70 | Relationships between algebraic curves and integrable systems |

53D18 | Generalized geometries (à la Hitchin) |

14H60 | Vector bundles on curves and their moduli |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |

14-04 | Software, source code, etc. for problems pertaining to algebraic geometry |

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\textit{O. K. Sheinman}, Dokl. Math. 97, No. 2, 144--146 (2018; Zbl 1432.14030); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 479, No. 3, 254--256 (2018)

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