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The sum of elliptic integrals simultaneously determines orbits in thr Keplerproblem and the addition of divisors on elliptic curves. Periodic motion of abody in physical space is defined by symmetries, whereas periodic motion ofdivisors is defined by a fixed point on the curve. Algebra of the firstintegrals associated with symmetries is a well-known mathematical object,whereas algebra of the first integrals associated with coordinates of fixedpoints is unknown. In this paper, we discuss polynomial algebras ofnon-polynomial first integrals of superintegrable systems associated withelliptic curves.

The Kepler problem: polynomial algebra of non-polynomial first integrals