Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 66 / 4 / i / 3

More than three integrals occur when additional symmetry or degeneracy restricts motion beyond a generic three-frequency invariant torus. In a time-independent central gravitational potential, energy and the three components of specific angular momentum are four generically independent isolating integrals. They fix the orbital plane as well as the radial-motion parameters. A nonclosed planar rosette typically has two-dimensional orbit closure, rather than a three-dimensional invariant torus.

The Kepler orbit is a stronger example. In addition to $E$ and $\mathbf{L}$, the conserved Laplace-Runge-Lenz vectorfixes the apsidal direction. Its constraints $\mathbf{A}\cdot \mathbf{L}=0$ and $A^2={\mu }^2+2E{L}^2$ leave five independent integrals among $E,\mathbf{L},\mathbf{A}$. It is a maximally superintegrable Hamiltonian system, with closed nondegenerate bound orbits.

More than three integrals need not form a set of Poisson-commuting functions; the three-integral limit applies to an independent commuting set, rather than to every conserved orbit constraint. At a nonstationary point in six-dimensional phase space, integral gradients annihilate the nonzero flow vector, so at most five are independent. This explains the isolating integrals and orbital closure dimension count. A resonance on one isolated orbit does not by itself prove an additional smooth global integral on a neighborhood.