Completely Integrable Lax Pair equations for Connes-Kreimer Hopf algebra of rooted trees

We show the complete integrability of the Lax pair equations for certain low-dimensional Lie algebras of the infinitesimal character ${\tilde{\beta }}_0$ introduced in the paper \emph{Lax pair equations and {C}onnes-{K}reimer renormalization}, by G. B{ă}di{ţ}oiu and S. Rosenberg, Comm. Math. Phys. $\mathbf{\text{296}}$ (2010), 655–680.