Calculation of Normal Forms of the Euler-Poisson Equations

In the paper [1], the special case of the Euler-Poisson equations describing movements of a heavy rigid body with a fixed point is considered. Among stationary points of the system, two of one-parameter families were chosen. These families correspond to the resonance of eigen-values (0, 0,lambda,-lambda, 2 lambda,-2 lambda) of the matrix of the linear part of the system, also in [1] it was conjectured the absence of the additional first integral (with respect to well-known 3 integrals (2)) near these families, except of classical cases of global integrability. In this paper, the supposition is proved by calculations of coefficients of the normal form.