The preprint consists of two sections. In § 1 we study an ODE system which describes an evolution of average values of coordinates and momentum of a quantum particle in the approximation Ο(h³/²) . The system is a simple case of one derived by V.Belov and V.Maslov. We construct the Poisson structure, with respect to which the system has the hamiltonian form. Then we investigate in detail a system of order 5 corresponding to one-dimensional unharmonic oscillator. Using the normal form method in the vicinity of the origin we integrate the first approximation of the system. In § 2 an analytical volume-preserving ODE system near a stationary point is considered. The system is assumed to be in m-multiple resonance and have m-2 known integrals. The systems from § 1 has these properties with m=4.The result is that such a system has another independent formal integral. We study the structure of this integral. The sections can be read independently and have separate numeration of formulas, theorems, propositions and remarks.

Mathematical problems and theory of numerical methods

Bruno Alexander Dmitrievich,  abruno@keldysh.ru,  orcid.org/0000-0002-7465-1258,  KIAM RAS

Sadov S. Y.,  KIAM RAS