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Well, more generally, a Hamiltonian system $\dot{z}^I=\{z^I,H\}$ with a Hamiltonian function $H:M\times \mathbb{R}\to \mathbb{R}$ is defined on a (not necessarily invertible) Poisson manifold rather than a symplectic manifold.

A (not necessarily invertible) Poisson manifold might not have local canonical/Darboux coordinates, cf. OP's example.

An important example of a non-invertible Poisson bracket is the Dirac bracket for constrained systems.