Scientific paper ID 1642 : 2018/3

Abstract. In this paper it is shown that the Hamiltonian system with Dyson potential is analytical non-integrable and formal non integrable. The Dyson system, appeared in 1962 by Dyson statistical physics research of the energy levels of one-dimensional Coulomb’s gas. There is connection between the of system of point vortices - this is result by Calogero and Perelomov in 1978. Hamiltonian system has two integrals, that is why for integrability it is important the first nontrivial case . Borisov and Kozlov in 1998 had proved that the system in case is non-integrable in analytical first integrals using splitting separarises method. In this paper it is proved the same statement, but way is different. Proof based of Duistermaat’s idea when a system has a family of periodic solutions around equilibrium, and if the period function is infinitely branched, then if the system has additional analytical first integral - this integral is a constant. We call that the system with Hamiltonian H formally integrable if there exist formal power series in involution, where are functionally independent and Taylor expansion of H is a formal power series in . An asymptotic behavior near equilibrium is like an integrable system. The formal integrability gives asymptotic information about the flow. For the proof of formal non-integrability in the paper is used Theory of Ziglin - Morales-Ruiz-Ramis. Computer algebra method is the Kovacic algorithm using a software for education, engineering and research Maple.