A theorem is derived which determines higher order first integrals of autonomous holonomic dynamical systems in a general space, provided the collineations and the Killing tensors –up to the order of the first integral– of the kinetic metric, defined by the kinetic energy of the system, can be computed. The theorem is applied in the case of Newtonian autonomous conservative dynamical systems of two degrees of freedom, where known and new integrable and superintegrable potentials that admit cubic first integrals are determined.

中文翻译：

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自主动力系统的高阶一阶积分

推导出一个定理，该定理确定一般空间中自主完整动力系统的高阶一阶积分，前提是由系统的动能定义的动力学度量的共线和杀戮张量 - 达到一阶积分的阶，可以计算。该定理适用于两个自由度的牛顿自治保守动力系统，其中确定了允许三次第一积分的已知和新的可积和超可积势。