= Solution
Write the <Hamiltonian> as the sum of separated one-dimensional energies
$$
H=\sum_{i=1}^n H_i,
\qquad
H_i=\frac12\left(p_i^2+\omega_i^2q_i^2\right).
$$
Each $(q_i,p_i)$ system is a <simple harmonic oscillator> and is integrable away from $(q_i,p_i)=(0,0)$ by part (b). The separated energies satisfy
$$
\{H_i,H_j\}=0,
$$
and their differentials are independent whenever every oscillator pair is nonzero. The <Poisson-commuting separated energies> therefore make the $n$-dimensional oscillator integrable. The exceptional coordinate strata have measure zero, so this proves <Liouville integrability> in its usual almost-everywhere sense; the origin is the only point where $dH$ itself vanishes.
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