Computation 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. In the family we select one-parameter families with fixed resonances 1:2 and 1:3. For them, we study the structure of the normal form and of the first integrals. By a computation of the normal form, we found that conditions, which are necessary for the existence of an additional local first integral, are violated in all cases, except classical cases of global integrability.