We build upon the previous chapter’s material and derive an equation satisfied by a canonical transformation that completely solves the system by bringing the new Hamiltonian to zero. This equation is called the Hamilton-Jacobi equation. It is a nonlinear partial differential equation; any solution of this equation provides a complete solution of the corresponding Hamilton’s equation. While the Hamilton-Jacobi equation is not solvable in the general case, we provide some cases, called separable systems, where a particular solution can be found. We further discuss the action-angle variables and state the Liouville-Arnol’d’s integrability theorem. We conclude the chapter by discussing the adiabatic integral for one-dimensional systems with the Hamiltonian slowly changing in time.