Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 3 / e / Solution

Axisymmetry supplies conserved $p_{\theta }$ in addition to conserved energy. Consequently $\boxed{f=F(H,p_{\theta })}$ is a more general stationary solution, since both arguments are constant on every trajectory. Additional independent integrals may permit still more general distributions. This is a sufficient construction, not a claim that every stationary distribution must have exactly these two arguments.