Darboux transformations and integrable differential–difference equations associated with Kac–Moody Lie algebras

It is well known that with every Kac–Moody Lie algebra one can associate an integrable two dimensional Toda type system. In paticular the sinh-Gordon equation corresponds to the algebra $A_1^{(1)}$, the Tzitzeica equation to $A_2^{(2)}$, the usual periodic Toda lattice to $A_n^{(1)}$, etc. In our work we construct integrable chains of B"acklund transformations for Toda type systems associated with the classical families of Kac–Moody algebras and derive Darboux transformations for the corresponding Lax operators. We also discuss integrable finite difference systems corresponding to the Bianchi permutability of the Bäcklund transformations.