Integrable systems on graphs (Q2780493)

The authors consider an integrable system of Krichever-Novikov type [\textit{I. M. Krichever} and \textit{S. P. Novikov}, Russ. Math. Surv. 54, No. 6, 1248-1249 (1999); translation from Usp. Mat. Nauk 54, No. 6, 149-150 (1999; Zbl 0981.37023)] on an arbitrary graph \(\Gamma\), and study some of its geometrical and dynamical properties. The authors prove that a real self-adjoint operator \(L\) of fourth order on \(\Gamma\) possesses an isospectral deformation of one energy level \(L\varphi=0\) in terms of an \((L,A,B)\)-triple, \(B=A_1\Delta T\). The deformation operator is given in terms of a cocycle.