We discuss the relationship between the singular points of an autonomous system of differential equations and the critical points of its first integrals. Applying the well-known Splitting Lemma, we introduce local coordinates in which the first integral takes a “canonical” form. These coordinates make it possible to introduce a quasihomogeneous structure in some neighbourhood of any singular point and so to prove general theorems on the existence of asymptotic trajectories which go into or out of that singular point. We consider quasihomogeneous truncations of the original system of differential equations and show that if the singular point is isolated, the quasihomogeneous system is Hamiltonian. For a general mechanical system with two degrees of freedom, we prove a theorem on the instability of an equilibrium when it is neither a local minimum nor a local maximum of the potential energy. Bibliography: 21 titles.

中文翻译：

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第一积分和渐近轨迹

我们讨论了微分方程自治系统的奇异点与其第一积分的临界点之间的关系。应用著名的分裂引理，我们引入第一个积分采用“规范”形式的局部坐标。这些坐标使得可以在任何奇异点的某个邻域中引入准均质结构，从而证明关于进入或离开该奇异点的渐近轨迹的存在的一般性定理。我们考虑了微分方程原始系统的拟均一截断，并表明，如果奇异点是孤立的，则拟均一系统是哈密顿量。对于具有两个自由度的通用机械系统，当它既不是势能的局部最小值也不是局部最大值时，我们证明了一个关于不稳定性的定理。参考书目：21种。