The 10 classical integrals of the three-body problem (three for the position of the center of mass, three for the velocity of the center of mass, three for the angular momentum, and one for the energy) are the only algebraically independent integrals of this 18-degree-of-freedom system.
Poincaré's Theorem, Three-Body Problem
References
Bruns, H. Ber. Königl. Sächs. Ges. der Wiss., pp. 1, 55, 1887.
Bruns, H. Acta Math. 11, 25.
Forsyth, A. R. Ch. 17 in Theory of Differential Equations, Vol. 3. New York: Dover, 1959.
Whittaker, E. T. "Bruns' Theorem." §164 in A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, pp. 358-379, 1944.
