I am reading Ordinary Differential Equations and Dynamical Systems by Gerald Teschl am confused on what is exactly meant by "completely integrable" in the following picture. This is Chapter 8 section 4. Also, they refer to the "Hamilton structure" in the text below. Does this mean the Hamilton's equations listed here: 
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A "completely integrable" Hamiltonian system is one that satisfies the Liouville-Arnold theorem. Building off what LutzL said, there exist $n$ independent first integrals.
If any of the integrals are dependent (nonvanishing under the Poisson bracket), then you need $>n$ first integrals. How many depends on the relationships given by the Poisson bracket.
"Canonical form" refers to the fact that the position and velocities are "on the same level" as each other, as opposed to the Lagrangian-style of dynamics. You'll have $2n$ first-order differential equations as opposed to the $n$ second-order differential equations from Lagrange. On a deeper level, this relates to the symplectic manifold that the Hamiltonian system lives in.