The sum of elliptic integrals simultaneously determines orbits in the Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. The algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas the algebra of the first integrals associated with the coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of nonpolynomial first integrals of superintegrable systems associated with elliptic curves.

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开普勒问题：非多项式第一积分的多项式代数

椭圆积分的总和同时确定开普勒问题中的轨道以及椭圆曲线上除数的增加。物体在物理空间中的周期性运动由对称性定义，而除数的周期性运动由曲线上的固定点定义。与对称性关联的第一积分的代数是众所周知的数学对象，而与固定点的坐标关联的第一积分的代数是未知的。在本文中，我们讨论了与椭圆曲线相关的超可积系统的非多项式第一积分的多项式代数。