Andrey Oshemkov
Lomonosov Moscow State University,GSP1, Leninskie Gory, Moscow, 119991, Russia
Department of Mathematics and Mechanics, M.V. Lomonosov Moscow State University
Publications:
, Oshemkov A. A., Sokolov S. V.
The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram
2016, vol. 21, no. 5, pp. 581592
Abstract
The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $LA$ pair found by Reyman and SemenovTianShansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler – van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to selfintersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler – van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on $so(4)$.

Bolsinov A. V., Oshemkov A. A.
BiHamiltonian structures and singularities of integrable systems
2009, vol. 14, nos. 45, pp. 431454
Abstract
A Hamiltonian system on a Poisson manifold $M$ is called integrable if it possesses sufficiently many commuting first integrals $f_1, \ldots f_s$ which are functionally independent on $M$ almost everywhere. We study the structure of the singular set $K$ where the differentials $df_1, \ldots, df_s$ become linearly dependent and show that in the case of biHamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the biHamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying biHamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.
