In the Hamiltonian formulation of classical mechanics, we have the result that, in a system with Hamiltonian $H(q_i,p_i),\ i=1,\dots,N$, a quantity $f(q_i,p_i)$ time-evolves according to $$ \frac{\mathrm{d} f}{\mathrm{d} t} = \{f,H\} = \frac{\partial f}{\partial q_i}\frac{\partial H}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial H}{\partial q_i} $$ and so $f$ is conserved if and only if it Poisson commutes with $H$, i.e. $\{f,H\} = 0$.
In the case $N=1$ so that the phase space is 2-dimensional, and contour surfaces of functions are curves. Contours of $H$ are trajectories, and $f$ is a conserved quantity if and only if the contours of $f$ align with those of $H$. But for $N\geq2$, the contour surfaces of $H$ and $f$ are $(2N-1)$-dimensional manifolds in the $2N$-dimensional phase space. What can you say about these surfaces when $\{f,H\} = 0$?
In quantum mechanics, when the functions $f$ and $H$ are replaced by operators $\hat{f}$ and $\hat{H}$ and the Poisson bracket is replaced by the commutator, then $[\hat{f},\hat{H}] = 0$ means that the two operators have coinciding eigenspaces. Is there such a nicely geometric way of visualising the classical equivalent?