Poincaré made essential use of Bruns's results in his famous memoir Le Probléme des Trois Corps [The Three Body Problem], Revue générale des sciences pures et appliquees 2, 1-5 (1891). Here is Poincare's summary of Bruns's results, and his own, translated by Chenciner in Poincaré and the Three-Body Problem:
"The differential equations of the Three-Body Problem possess a number of integrals which have long been familiar; these are those of the motion of the center of mass, the area integrals, the energy. It was extremely unlikely that they could have other algebraic integrals; it is, however only in the recent years that Mr. Bruns has proved this rigorously. But we can go further; apart from the known integrals, the Three-Body Problem admits no analytic and uniform integral; a careful study of the properties of periodic and asymptotic solutions is enough to establish this. It can be concluded that the various developments proposed so far are divergent; for their convergence would imply the existence of a uniform integral."
Chenciner gives a comprehensive discussion, and references Bruns's original paper, H. Bruns, Über die Integrale des Vielkörper-problems [On the Integrals of the Multi-body Problems], Acta Mathematica,
volume 11 (1887). Also relevant is Poincare's paper Sur la Méthode de Bruns [On Bruns's method], C.R.A.S. 1896, t. 123, 1224-1228.
Curiously enough, despite all this there exists a convergent power series solution to the 3-body problem, which was found by Sundman in 1913. Saari gives an accessible account of Sundman's construction in A Visit to the Newtonian N-body Problem via Elementary Complex Variables:
"Ironically, one of his major conclusions killed interest in a line of inquiry, so this particular result is not very well known. It should be; it is where Sundman "solved" the three-body problem according to accepted standards of the late 1800s and early 1900s.Indeed, in the late 1800s the King of Sweden and Norway established a prize for anyone who could find the solution of the N-body problem. The prize was awarded to Poincare in 1889 even though he hadn't solved the original problem. (On the other hand, Poincare's prizewinning paper contains a wealth of ideas that remain influential.) The originally stated problem finally was solved in 1913 by Sundman [16] when he found a converging series solution for the three-body problem. Unfortunately, his series converges so slowly that, essentially, it is useless for any practical purpose."
This is a rare example of an explicitly constructive solution coming so far apart from practically useful. Nonetheless, there is a renewed interest recently in generalizing Sundman's methods to the $N$-body problems with $N>3$.
