Consider a six-dimensional phase space with coordinates $\left(q_{i}, p_{i}\right)$ for $i=1,2,3$. Compute the Poisson brackets $\left\{L_{i}, L_{j}\right\}$, where $L_{i}=\epsilon_{i j k} q_{j} p_{k}$.

Consider the Hamiltonian

$$H=\frac{1}{2}|\mathbf{p}|^{2}+V(|\mathbf{q}|)$$

and show that the resulting Hamiltonian system admits three Poisson-commuting independent first integrals.