Context. I am reading a physics book in which it shown that, in the Hamiltonian setting, a bijective coordinate transformation
$$ q \rightarrow Q \text{ s.t. }Q = Q(q,p)\\ p \rightarrow P \text{ s.t. }P = P(q, p) $$
preserves the Hamilton equations
$$ \frac{dQ}{dt} = \frac{\partial \tilde{H}}{\partial P} \\ \frac{dP}{dt} = \frac{\partial \tilde{H}}{\partial Q} $$
where $\tilde{H} = H(Q^{-1}(Q,P), P^{-1}(Q,P))$, if and only if the Poisson bracket of $P$ and $Q$ satisfies
$$ \{Q, P\}_{q, p} = \left( \frac{\partial Q}{\partial q} \frac{\partial P}{\partial p} - \frac{\partial Q}{\partial p} \frac{\partial P}{\partial q}\right )\ne 0 $$
It is apparently also usually desired that $\{Q, P\}_{q, p} = 1$.
Question: Is there a geometric interpretation (or some other sort of intuitive explanation) as to what the condition $\{Q, P\}_{q, p} = 0$ means, and why it destroys the Hamiltonian equations?