= Isolating integrals and orbital closure dimension
{title2=$N=6-d$}
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.
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