Stability Criteria of Equilibrium Resonance Position in Systems Admitting First Integral

Systems of smooth differential equations in $\mathbb{R}^4$ are considered, which possess the first integral and for which the origin is a nondegenerate equilibrium position. It is assumed that the linear part of such systems has two pairs of pure imaginary eigenvalues $\pm i\omega_1$, $\pm i \omega_2$. For the given two-frequency problem the stability and instability criteria are istablished in a case when the frequences $\omega_1$ and $\omega_2$ are incommensurable as well as in a case of different resonance correlations between them. These criteria are based on the shape of Poincaré-Dulac normal form of corresponding equations of not more than the third order.