On the Differential equations of the characters for the Renormalization   group
  Owing to the analogy between the Connes-Kreimer theory of the renormalization and the integrable systems, we derive the differential equations of the unit mass for the renormalized character $\phi_+$ and the counter term $\phi_-$. We give another proof for the scattering type formula of $\phi_-$. The differential equation of the counter term of coordinate on ${\mathbb P}^1$ is also given. The hierarchy of the renormalization groups is defined as the integrable systems. 
Masato Sakakibara