= Solution
Write $H=F(P,Q)$ with $P=\sum p_k^2$ and $Q=\sum q_k^2$. The <angular momentum> component $L_i=\epsilon_{ijk}q_jp_k$ has $\partial_{p_\ell}L_i=\epsilon_{ij\ell}q_j$ and $\partial_{q_\ell}L_i=\epsilon_{i\ell k}p_k$. Therefore
$$
\{H,L_i\}=2F_Q\sum_{j,\ell}\epsilon_{ij\ell}q_\ell q_j
-2F_P\sum_{\ell,k}\epsilon_{i\ell k}p_\ell p_k=\boxed{0},
$$
since a symmetric product contracts to zero with the antisymmetric <Levi-Civita symbol>. All three components of <angular momentum> are conserved by this rotationally invariant <Hamiltonian>.
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