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

For one degree of freedom, take $F_1=H$. Antisymmetry of the Poisson bracket gives $\{H,H\}=0$, so energy is conserved. The stated condition is exactly $dH\ne 0$, which supplies the required functional independence. Hence the system is integrable at every such regular point.