We study the integrability of the Hamiltonian normal form of 1:2:2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains many parameters. For a generic choice of parameters in the normal form up to order four, we carry on non-integrability analysis, based on the Morales–Ramis theory using only first variational equations along certain particular solutions. The non-integrability obtained by algebraic proofs produces dynamics illustrated by some numerical experiments.We also isolate a non-trivial case of integrability.

中文翻译：

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关于哈密顿1：2：2共振的可积性

我们研究了1：2：2共振哈密顿正态形式的可积性。众所周知，这种截断为三阶的范式是可积的。被截断的四阶范式包含许多参数。对于一般形式的参数（最多四阶），我们基于莫拉莱斯-拉米斯理论，仅使用某些特定解的一阶变分方程，进行非可积​​性分析。通过代数证明获得的非可积性产生了一些数值实验说明的动力学。我们还分离出了一个非平凡的可积性案例。