= Solution
The Hamiltonian in the question is
$$
H=\frac12\bar z_jz_j
=\frac12\sum_{j=1}^n(q_j^2+p_j^2).
$$
Define
$$
I_j=\frac12|z_j|^2=\frac12(q_j^2+p_j^2).
$$
Each $I_j$ is a first integral, the functions involve disjoint canonical coordinate pairs and hence satisfy $\{I_j,I_k\}=0$, and their differentials are independent wherever every $z_j\ne0$. Since $H=\sum_jI_j$, these $n$ integrals prove <Liouville integrability of the isotropic harmonic oscillator>.
For constants $c_j>0$, the common level is
$$
I_j=c_j
\quad\Longleftrightarrow\quad
|z_j|=\sqrt{2c_j},
$$
so the regular invariant set is
$$
\boxed{\ S^1_{\sqrt{2c_1}}\times\cdots\times S^1_{\sqrt{2c_n}}\cong\mathbb T^n.\ }
$$
Hamilton's equations give $\dot z_j=-iz_j$, so every angle evolves as $\theta_j(t)=\theta_j(0)-t$. If some $c_j=0$, the corresponding circle collapses and the level is a lower-dimensional singular invariant torus.
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