Normal Forms and Integrability of the Euler–Poisson Equations

For a system of ordinary differential equations, we introduce a concept of its local integrability near a stationary solution and near a power asymptotics of its solution. For the local analysis of the system near its stationary solution, we propose to compute its normal form. The power asymptotics of its solution can be translated into a stationary solution by means of a power transformation of coordinates and then we can use the normal form. The approach is applied for some solutions to a particular case of the Euler–Poisson system of equations, describing motion of a rigid body with a fixed point. We have found that as a rule the system is locally nonintegrable near its stationary solutions and is locally integrable near power asymptotics of its solutions.