Understanding what is meant by "Integrability" with regards to the Euler Top I am reading Discrete Systems and Integrability by F.W. Nijhoff, J. Hietarinta, and N. Joshi. Currently, I am investigating Chapter 6 and in particular the Euler Top. The book says the following:

There are several properties associated with integrability of maps,
  for example, the existence of a sifficient number of conserved
  quantities, symmetries, Lax pair and the behavior around
  singularities.

The book then defines a type of Integrability

A system with $2N$-dimension phase space is called Louiville
  Integrable if there are $N$ conserved quantities

The Euler Top is  given by
$$
\begin{cases}
\overset{\cdot}{x}_1 = \alpha_1 x_2 x_3, \\
\overset{\cdot}{x}_2 = \alpha_2 x_{3}x_{1}, \\
\overset{\cdot}{x}_3 = \alpha_3 x_{1}x_{2}.
\end{cases}
$$
where $\alpha_{1,2,3}$ are real parameters. This is one of the most famous integrable systems of classicaly mechanics. The function $H(x) = \gamma_1 x_1^2 + \gamma_2 x_2^2 + \gamma_3 x_3^2$ is an integral (conserved quantity?) for the above system $\iff \gamma \perp \alpha$.
In particular, there are $3$ integrals of motion (conserved quantities) of the system where only two of them are independent:
$$H_1 = \alpha_2 x_3^2 - \alpha_3 x_2^2, \; H_2 = \alpha_3 x_1^2 - \alpha_1 x_3^2 , \; H_3 = \alpha_1 x_2^2 - \alpha_2 x_1^2$$
So, here is what I know. The dimension of the Euler Top is $3$ because there are three independent variables $x_1, x_2, x_3$. We have found $2$ independent conserved quantities. Since the dimension of the system is $3$, then this doesn't fit the definition of the Louiville Integrability. 
So, by what definition is the Euler Top called Integrable? 
 A: Hamiltonian systems can be defined on any manifold where there is a Poisson bracket (something satsifying the axioms given in section 6.1.1 of the book you referred to). It doesn't have to be a $2N$-dimensional space, and the Poisson bracket doesn't have to be given by the classical formula
$$
\{ F,G \} = \sum \left( \frac{\partial F}{\partial q_j} \frac{\partial G}{\partial p_j} - \frac{\partial F}{\partial p_j} \frac{\partial G}{\partial q_j} \right)
.
$$
In the case of the Euler rigid body equations with no external forces, which are usually written
$$
\begin{aligned}
  \dot x_1 = \left( \frac{1}{I_3} - \frac{1}{I_2} \right) x_2 x_3, \\
  \dot x_2 = \left( \frac{1}{I_1} - \frac{1}{I_3} \right) x_1 x_3, \\
  \dot x_3 = \left( \frac{1}{I_2} - \frac{1}{I_1} \right) x_1 x_2,
\end{aligned}
$$
where $(x_1,x_2,x_3)=(I_1 \omega_1,I_2 \omega_2,I_3 \omega_3)$,
the phase space is $\mathbf{R}^3$ and the Poisson bracket is
$$
\{ F,G \} =
\begin{pmatrix}
  \dfrac{\partial F}{\partial x_1} &
  \dfrac{\partial F}{\partial x_2} &
  \dfrac{\partial F}{\partial x_3}
\end{pmatrix}
\begin{pmatrix}
  0 & -x_3 & x_2 \\
  x_3 & 0 & -x_1 \\
  -x_2 & x_1 & 0
\end{pmatrix}
\begin{pmatrix}
  \partial G / \partial x_1 \\
  \partial G / \partial x_2 \\
  \partial G / \partial x_3
\end{pmatrix}
.
$$
The matrix in the middle is called a Poisson matrix, and the fact that this really is a Poisson bracket (in particular that the Jacobi identity is satisfied) has to do with the Lie algebra $\mathfrak{so}(3)$.
The Euler top is a Hamiltonian system (with respect to this Poisson structure) since it can be written as
$$
\frac{d}{dt}
\begin{pmatrix}
  x_1 \\ x_2 \\ x_3
\end{pmatrix}
=
\begin{pmatrix}
  0 & -x_3 & x_2 \\
  x_3 & 0 & -x_1 \\
  -x_2 & x_1 & 0
\end{pmatrix}
\begin{pmatrix}
  \partial H / \partial x_1 \\
  \partial H / \partial x_2 \\
  \partial H / \partial x_3
\end{pmatrix}
,
$$
with the Hamiltonian function
$$
H(x_1,x_2,x_3) = \frac12 \left( \frac{x_1^2}{I_1} + \frac{x_2^2}{I_2} + \frac{x_3^2}{I_3} \right)
.
$$
This automatically makes $H$ a constant of motion.
And moreover, $C=x_1^2+x_2^2+x_3^2$ is another constant of motion. It's a so-called Casimir function, which means that it's constant for any system of the above Hamiltonian form, no matter what the Hamiltonian function $H$ is (so it's something that's only related to the particular Poisson structure that we have here). And it turns out that you can restrict the Poisson structure to each level set of $C$ (in this case spheres around the origin, and let's exclude the exceptional level set consisting of only the origin), so that each sphere becomes a symplectic manifold, which is the setting for classical Hamiltonian systems. And the system can be restricted to a system on any particular such “symplectic leaf”, as these spheres are called, and you can introduce coordinates $(q,p)$ such that the system looks just like a classical Hamiltonian system (in this case $2N$-dimensional with $N=1$). And any such system is Liouville integrable, since it has the right number of constants of motion: one (namely the Hamiltonian function, the restriction of $H$ to the sphere).
This was of course only a very brief outline. A more thorough description of how it works in general can be found in Chapter 6 of Peter Olver's Applications of Lie Groups to Differential Equations.
