= Solution
Use the canonical <Poisson bracket> convention $\{H,G\}=\sum_k(H_{q_k}G_{p_k}-H_{p_k}G_{q_k})$. The <Hamiltonian flow> generated by $G$ has $\dot q_k=G_{p_k}$ and $\dot p_k=-G_{q_k}$. The <chain rule> therefore gives
$$
\frac{dH}{dt}=\sum_k(H_{q_k}\dot q_k+H_{p_k}\dot p_k)=\{H,G\}.
$$
Thus $H$ is invariant under every such trajectory exactly when $\boxed{\{H,G\}=0}$ identically. Antisymmetry then gives $dG/dt=\{G,H\}=0$ along the flow of $H$, so $G$ is a <constant of motion>.
For $G=\alpha\sum_kp_k$, the generated equations are $\dot q_k=\alpha$, $\dot p_k=0$, hence $q_k(t)=q_k(0)+\alpha t$, $p_k(t)=p_k(0)$. Invariance under this simultaneous translation implies conservation of $\boxed{\sum_kp_k}$ when $\alpha\ne0$; the zero generator alone conveys no additional conservation law.
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