We consider systems of differential equations with equilibrium points. These systems are formally transformed to Poincaré-Dulac normal forms, but the normalization may not be analytic. Such an analytic normalization is a classical problem that dates back to Poincaré. Zung  proved that analytically integrable systems of differential equations have analytic normalizations. Nonintegrability and nonexistence of analytic normalizations are equivalent when the resonance degree is less than two since their Poincaré-Dulac normal forms are integrable . In this talk, we prove that a special class of two-dimensional systems whose resonance degree is one have no analytic normalizations so that they are not integrable.
 S. Yamanaka, Local integrability of Poincaré-Dulac normal forms, Regul. Chaotic. Dyn. 23 (2018), 933–947.
 N. T. Zung, Convergence versus integrability in Poincaré-Dulac normal forms, Math. Res. Lett. 9 (2002), 217–228.