In this paper we study analytically the existence of two families of periodic orbits using the averaging theory of second order, and the finite and infinite equilibria of a generalized Hénon-Heiles Hamiltonian system which includes the classical Hénon-Heiles Hamiltonian. Moreover we show that this generalized Hénon-Heiles Hamiltonian system is not $C^1$ integrable in the sense of Liouville–Arnol'd, i.e. it has not a second $C^1$ first integral independent with the Hamiltonian. The techniques that we use for obtaining analytically the periodic orbits and the non $C^1$ Liouville-Arnol'd integrability, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

在本文中，我们使用二阶平均理论以及包括经典Hénon-Heiles哈密顿量的广义Hénon-Heiles哈密顿量系统的有限和无限均衡，分析了两个周期轨道族的存在。此外，我们证明了这种广义的Hénon-Heiles哈密顿系统不是$C^{\mathrm{1个}}$ 在Liouville–Arnol'd的意义上是可集成的，即它没有第二个 $C^{\mathrm{1个}}$独立于哈密顿量的第一个积分。我们用于解析获得周期轨道和非轨道轨道的技术$C^{\mathrm{1个}}$ Liouville-Arnol的可积性可以应用于具有任意数量的自由度的哈密顿系统。