Normal forms and integrability of ODE systems

Abstract

We consider a special case of the Euler-Poisson system describing the motion of a rigid body with a fixed point. This is an autonomous sixth-order ODE system with one parameter. Among the stationary points of the system, we select two one-parameter families with resonance (0, 0, λ, −λ, 2λ, −2λ) of eigenvalues of the matrix of the linear part. At these stationary points, we compute the resonant normal form of the system using a program based on the MATHEMATICA package. Our results show that, if there exists an additional first integral of the system, then its normal form is degenerate. Therefore, we assume that the integrability of the ODE system can be established based on its normal form.