 Isolating integrals and orbital closure dimension

In three-dimensional autonomous mechanics, a regular common level set of $N$ independent isolating integrals has dimension $6-N$ in phase space. If an orbit is dense in a $d$-dimensional invariant torus and the integrals characterize its closure, $N=6-d$. A generic three-frequency regular orbit has three integrals; a spherical nonclosed rosette has energy and three components of specific angular momentum, with two-dimensional closure and four integrals. A periodic nonstationary orbit has one-dimensional closure and can be isolated by five independent orbit labels on the invariant family. The maximum is five wherever the Hamiltonian vector field is nonzero, since all integral differentials annihilate that field. A single resonant orbit need not yield an additional smooth single-valued integral on an open neighborhood; global superintegrability is stronger than orbital closure.