  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{\u{a}}di{\c{t}}oiu and S. Rosenberg, Comm. Math.
Phys. \textbf{296} (2010), 655--680.
