Invariant and integral manifolds of dynamical systems and the problem of integration of the Euler–Poisson equations

The problem on inclusion of an invariant manifold of a dynamical system into a set of integral manifolds is considered. It is shown that this inclusion is always possible unless the invariant manifold is a singular one (consisting of singular points of the system) of codimension one. It enables us to study invariant manifolds using the equation for integrals instead of the Levi-Civita equations containing uncertain factors. The defining role of the singular manifolds in shaping a phase portrait of a dynamical system is established and the following from this integrals properties are obtained. The results are applied to the analysis of solutions of the Euler–Poisson equations. A new treatment of the fourth integrals in Euler's, Lagrange's, Kovalevskaya's cases is proposed. It is proved that under the Hess conditions there is a fourth integral of which special cases are Euler's and Lagrange's integrals as well as the Hess and the Dokshevich solutions.