The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

中文翻译：

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受限三体问题的不可积性

圆形受限三体问题的不可积问题是动力系统理论中非常经典和重要的问题。庞加莱在十九世纪部分解决了这个问题：他证明不存在实解析第一积分，它在分析上取决于第二物体与总质量的比率，并且在功能上独立于哈密顿量。当第二个物体的质量变为零时，受限三体问题就简化为双体开普勒问题。我们证明了受限三体问题在平面和空间情况下对于第二体的任何非零质量的不可积性。我们证明的基本工具是这里开发的一种技术，用于确定可能是非哈密尔顿的可积系统的扰动是否在共振周期轨道附近不可亚纯可积，使得第一积分和交换向量场也亚纯地依赖于扰动参数。该技术基于 Ayoul 和 Zung 的莫拉莱斯-拉米斯和莫拉莱斯-拉米斯-西莫理论的广义版本。我们强调，我们的结果不仅仅是理论的应用。