Normal forms and integrability of ODE systems (Q2433011)

The connection between normal forms and integrability of a system is studied. For this purpose normal forms of the Euler-Poisson equations are computed. These equations in the case under consideration have the form \[ \begin{cases}\dot{p}=(1-c)qr,\\ \dot{q}=(c-1)pr-\gamma_3,\\ \dot{r}=\gamma_2/c, \end{cases}\tag{1} \] \[ \begin{cases} \dot{\gamma}_1=r\gamma_2-q\gamma_3,\\ \dot{\gamma}_2=p\gamma_3-r\gamma_1,\\ \dot{\gamma}_3=q\gamma_1-p\gamma_2, \end{cases}\tag{2} \] and describe the motion of a rigid body with a fixed point, \(c \in (0,2]\). The system (1)--(2) has three first integrals and a two-parameter family of stationary points. Among these points two one-parameter families with eigenvalues \( 0,0,\lambda,-\lambda,2\lambda,-2\lambda\) are selected. For stationary points of these families the normal form of the system up to the fifth order are determined. This normal form appears to be degenerate in known integrable cases and nondegenerate in known nonintegrable cases. If the considered system has an additional global first integral, then its normal form is degenerate for both families of stationary points. Otherwise, the normal form has no such degeneracy. It is either nondegenerate for both families or degenerate for only one family. There are four cases of this degeneracy. In this four cases the system has local first integrals near the values of the parameter where the normal form becomes degenerate. The resonance \(1:3\) is also analyzed. A more careful analysis shows that, in all four cases, the normalized system has no additional formal integral, i.e., is nonintegrable. It turns out that, in these cases, the normalized system has an additional family of periodic solutions which are close to the stationary point. It is shown that there are stationary points in whose neighborhoods the system is locally integrable.