We consider the planar three-body problem with generalized potentials. Some nonintegrability results for these systems have been obtained by analyzing the variational equations along homothetic solutions. But we cannot apply it to several exceptional cases. For example, in the case of inverse-square potentials, the variational equations along homothetic solutions are solvable. We obtain sufficient conditions for nonintegrability for these exceptional cases by focusing on some particular solutions that are different from homothetic solutions.

中文翻译：

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广义力简化平面三体问题的不可积性

我们考虑具有广义势的平面三体问题。通过分析沿齐次解的变分方程，已经获得了这些系统的一些不可积性结果。但我们不能将其应用于几个例外情况。例如，在反平方势的情况下，沿着齐次解的变分方程是可解的。我们通过关注一些不同于相似解的特定解，为这些例外情况获得了不可积性的充分条件。