Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 32C / c / Solution

Let $F_1,\dots ,F_n$ be commuting independent integrals for $(M,H)$ and let $G_1,\dots ,G_m$ be such integrals for $(\tilde{M},\tilde{H})$. Regard each as a function on the product phase space. ThenThe $F_i$ commute among themselves, the $G_a$ commute among themselves, and every cross bracket $\{F_i,G_a\}$ vanishes because the functions depend on disjoint canonical coordinates. Their differentials form two independent coordinate blocks, so all $n+m$ are independent on the product of the two regular sets. Thus the combined system is integrable, an instance of the direct product of integrable Hamiltonian systems.