Under study are the problems of existence of the first integrals rational in momenta for a natural mechanical system on the 2-torus. We prove that a rational integral with a linear numerator and denominator reduces to a linear integral. Considering the case of a quadratic numerator and linear denominator, we obtain an analogous result under some additional assumptions on the coefficients of the first integral.

中文翻译：

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2-圆环上自然机械系统的有理积分

正在研究的是在 2 环面上的自然机械系统的动量中有理第一积分的存在问题。我们证明了具有线性分子和分母的有理积分简化为线性积分。考虑二次分子和线性分母的情况，我们在对第一次积分系数的一些附加假设下获得了类似的结果。