Theory of normal forms of the Euler-Poisson equations

We consider the special case $A=B$, $x_0\ne0$, $y_0=z_0=0$ of the Euler-Poisson system of equations, describing the motion of a rigid body with a fixed point. Near a two-parameter family of its stationary solutions we study its normal forms. Using them, in the family we find three sets of stationary solutions, near which the system is locally integrable. One of these sets is real. Out the sets we select one-parameter families with a fixed resonance. For them, we study the structure of the normal form and of the first integrals. We point out conditions, which are necessary for the existence of an additional first integral.