Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb{R}^2$ in extended neighborhoods of simple singular points. I

Integrable Hamiltonian systems and Poisson actions of the group $\mathbb{R}^2$ with simple singular points on a smooth ($ C^\infty$ or real-analytic) four-dimensional symplectic manifold $(M,\,\Omega)$ are studied, where $\Omega$ is a symplectic 2-form.