= Solution
On a $2n$-dimensional <symplectic manifold>, a <completely integrable Hamiltonian system> has $n$ <first integrals in involution> $F_1=H,F_2,\ldots,F_n$ whose <differentials> are independent almost everywhere, hence on a dense open regular set:
$$
\boxed{\{F_i,F_j\}=0,\qquad dF_1\wedge\cdots\wedge dF_n\ne0.}
$$
An equivalent formulation allows $H$ to be a function of the $n$ commuting independent integrals. The independence requirement is on the regular set, permitting exceptional critical points and singular fibres.
For example, on $\mathbb R^{2n}$ take $H=\sum_{j=1}^nE_j$ with $E_j=(p_j^2+\alpha_j^2q_j^2)/2$ and $\alpha_j>0$. The disjoint coordinate pairs make all <Poisson brackets> $\{E_i,E_j\}$ zero, and the <differentials> are independent whenever no pair $(q_j,p_j)$ vanishes. Replacing $E_1$ by $H$ is an invertible <linear map> of this list. Thus $F_1=H,F_j=E_j$ for $j\ge2$ give the required example. Irrational frequency ratios affect orbit density, but not <complete integrability>.
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