I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of elementary functions. Is this correct? I get that for each linearly independent constant of motion you can reduce the degree of freedom by one, but I don't understand why the theorem
Theorem (First integrals of the n-body problem) The only linearly independent integrals of the $n$-body problem, which are algebraic with respect to $q$, $p$ and $t$ are the $10$ described above. (http://en.wikipedia.org/wiki/N-body_problem#Three-body_problem)
implies that there is no analytic solution (I think this is synonymous with closed-form solution, and solution in terms of elementary functions). I've been trying to think about it, but I can't reason it, and apparently integrability implies no chaos, which I can't see either.
2-Body Problems also exist which have no specific solution such that there is a range of solutions for a given physical condition. This means solvability is not based on the number of bodies but the state and representation of space.
Indian Journal of Science and Technology published a physical proof called, “Binary Precession Solutions based on Synchronized Field Couplings”
http://www.indjst.org/index.php/indjst/article/view/30008/25962
In this research, a generalized wave function with classical characteristics was isolated within the motion of binary stars. The wave function provided the first tool for cracking the complex motion of DI Herculis and other binary stars that had several measured precession solutions.
http://xxx.lanl.gov/pdf/1111.3328v2.pdf
In this research, published about a year after the Indian Journal of Science and Technology publication, mathematicians from Imperial College London produced a proof for the physical existence of wave functions. The research was published in Nature Magazine.