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?
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.