Cartan matrices and integrable lattice Toda field equations

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_{2},B_{2},C_{2},G_{2}$ the complete sets of integrals in both directions are found.
For the simple Lie algebras of the classical series $A_{N},B_{N},C_{N}$ and affine algebras of series $D_{N}^{(2)}$ the corresponding systems are supplied with the Lax representation.


References.
[1] Ismagil Habibullin, Kostyantyn Zheltukhin and Marina Yangubaeva, Cartan matrices
and integrable lattice Toda field equations, J. Phys. A: Math. Theor. 44 (2011) 465202
(20pp).
[2] Ismagil Habibullin, Rustem N. Garifullin, Affine Lie algebras and integrable Toda field
equations on discrete space-time, arXiv:1109.1689 (2011).