Find Lax pair for solid body equations

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Consider the system $$ \begin{align*} \dot{u_1}&=u_2u_3\\ \dot{u_2}&=u_3u_1\\ \dot{u_3}&=u_1u_2. \end{align*} $$

In the book "Integrable systems" by Hitchin et al., it is said, on p. 3, that

The point where most discussions of integrability begin is with the idea of a system of differential equations which can be put in Lax pair form. Let's begin with a finite-dimensional system $$ \frac{dA}{dt}=[A,B] $$ where $$ A(z)=A_0+zA_1+\cdots +z^nA_n,\quad B(z)=B_0+zB_1+\cdots + z^m B_m $$ are polynomials of $k\times k$ matrices.

Unfortunately, the book does not say what $[A,B]$ is supposed to mean. From asking the internet, I guess $[A,B]=AB-BA$.

Then, on p. 5, the book says:

Many finite-dimensional integrable systems fit into this scheme, and in particular the rigid body, where we can take $$ A(z)=\begin{pmatrix}0 & (u_1+u_2)\\(u_2-u_1) & 0\end{pmatrix}+z\begin{pmatrix}-2u_3 & 0\\ 0 & 2u_3\end{pmatrix}+z^2\begin{pmatrix}0 & (u_1-u_2)\\-(u_1+u_2) & 0\end{pmatrix}. $$

Here are some points which I do not understand.

  • What is $z$ here (and does it depend on $t$)? It seems to be some parameter but I do not understand the role of it.

  • Why is $A$ apparently $2\times 2$?

  • What is $B(z)$ here in order to get the Lex pair for the system above?


Here are the pages of the book I am referring to:

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