If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system
\begin{align*}L\psi&=\lambda\psi\\ \psi_t&=B\psi\end{align*}
has as its integrability condition the equation $$L_t=[B,L].$$
It then follows that the equation defined by the above is completely integrable, and what have you, because you can prove the existence of infinitely many first integrals in involution.
But this also makes the "equation of motion" for $L$, if we can call it that, look very similar to the dynamics of a quantity in a Hamiltonian system, where $B$ takes the role of a Hamiltonian $H$; just compare this with the equation of motion for any quantity $F$ with respect to time $t$:
$$\dot{F}=\{F,H\}+\frac{\partial F}{\partial t}$$
which when $F$ does not depend explicitly on $t$ is just the equation
$$\dot{F}=\{F,H\}.$$
Then this is suggestively similar to the behaviour of the Lax pair, with the Poisson bracket in the role of the commutator, and the operator $B$ in the role of the Hamiltonian, up to sign change.
Is this just a mundane mathematical coincidence, or is there some form of significance to the Lax operator formalism that makes it appear so similar to a Hamiltonian?
