1. Well, more generally, a Hamiltonian system is defined on a (not necessarily invertible) Poisson manifold rather than a symplectic manifold.

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

3. An important example of a non-invertible Poisson bracket is the Dirac bracket.

1. 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.

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

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

Well, more generally, a Hamiltonian system is defined on a (not necessarily invertible) Poisson manifold rather than a symplectic manifold. An important example of a non-invertible Poisson bracket is the Dirac bracket.

1. Well, more generally, a Hamiltonian system is defined on a (not necessarily invertible) Poisson manifold rather than a symplectic manifold.

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

3. An important example of a non-invertible Poisson bracket is the Dirac bracket.