Trivalent Graphs and Solitons

It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear systems representible through the ''L-A-B-triple''. The Laplace transformations for these operators are also constructed. Nothing like that exists for the second order difference operators on this tree.