Local integrability of the Euler-Poisson equations

The local integrity of Euler-Poisson equations describing the motion of a heavy rigid body with a fixed point is described. The normal forms of the system near two two-parameter families of its fixed solutions are studied. By using these normal forms in the families, 12 complex sets of fixed solutions are singled out near which the system is locally integrable. Both the global integrability of the system in the entire phase space and the local integrability of the system near its singular points can be analyzed. The local integrability of a system of ODEs near a singular point can be most easily established by analyzing its normal form. The properties of normal forms and normalizing transformations can also be found in the number of linearly independent solutions, which is called the multiplicity of a resonance.