  The sum of elliptic integrals simultaneously determines orbits in thr 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. Algebra of the first
integrals associated with symmetries is a well-known mathematical object,
whereas algebra of the first integrals associated with coordinates of fixed
points is unknown. In this paper, we discuss polynomial algebras of
non-polynomial first integrals of superintegrable systems associated with
elliptic curves.
