A geometric derivation of KdV-type hierarchies from root systems

For the root system of each complex semi-simple Lie algebra of rank two, and for the associated 2D Toda chain $E=\{u_{xy}=\exp (Ku)\}$, we calculate the two first integrals of the characteristic equation $D_y(w)=0$ on $E$. Using the integrals, we reconstruct and make coordinate-independent the $(2\times 2)$-matrixoperators$\square$ in total derivatives that factor symmetries of the chains. Writing other factorizations that involve the operators $\mathrm{\square }$, we obtain pairs of compatible Hamiltonian operators that produce KdV-type hierarchies of symmetries for $\text{cE}$. Having thus reduced the problem to the Hamiltonian case, we calculate the Lie-type brackets, transferred from the commutators of the symmetries in the images of the operators $\mathrm{\square }$ onto their domains. With all this, we describe the generators and derive all the commutation relations in the symmetry algebras of the 2D Toda chains, which serve here as an illustration for a much more general algebraic and geometric set-up.