A method for obtaining first integrals and integrating factors of autonomous systems and application to Euler-Poisson equations (Q2644316)

In the present paper the method of first integrals searching is proposed for \(n\)-th order autonomous systems. The technique is based on invariant manifolds theory. To describe the phase flow on an invariant manifold, the Levi-Civita equation [\textit{T. Levi-Civita} and \textit{U. Amaldi}, Lezioni di Meccanica Razionale. V.2. Bologna: Zanichelli, parte X (1952; Zbl 0047.17302)] is used. The starting point of the authors' investigation is the following paper [\textit{A. M. Kovalev}, Regul. Chaotic Dyn. 9, No. 1, 59--72 (2004; Zbl 1051.34038)]. As the technique application, several examples are considered. In particular the classical integral of the Kovalevskaya case is obtained. The authors assert that under the assumption \(A=B=2C\) choice of principal inertia axes can convert classical Kovalevskaya integral to the generalized form. Of course, this conclusion was known to S. Kovalevskaya too.