1. Well, more generally, a [Hamiltonian system](https://en.wikipedia.org/wiki/Hamiltonian_system) is defined on a (not necessarily invertible) [Poisson manifold](https://en.wikipedia.org/wiki/Poisson_manifold) rather than a [symplectic manifold](https://en.wikipedia.org/wiki/Symplectic_manifold). 

2. A (not necessarily invertible) Poisson manifold might not have locally [canonical](https://en.wikipedia.org/wiki/Canonical_coordinates)/[Darboux](https://en.wikipedia.org/wiki/Darboux%27s_theorem) coordinates, cf. OP's example. 

3. An important example of a non-invertible Poisson bracket is the [Dirac bracket](https://en.wikipedia.org/wiki/Dirac_bracket).