Normal Forms and Integrability of ODE Systems
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We consider a special case of the Euler–Poisson system describing the motion of a rigid body with a fixed point. It is the autonomous ODE system of sixth order 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. For the stationary points, we compute the resonant normal form of the system using a program based on the MATHEMATICA package. Our results show that in cases of the existence of an additional first integral of the system its normal form is degenerate. So we assume that the integrability of a system can be checked through its normal form.
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