Calculation of multiloop Feynman diagrams and diagonalization of commuting operators

We consider calculation of two families of multiloop Feynman diagrams: zig-zag diagrams and Basso-Dixon diagrams. The first family gives contribution to the beta-function of standard $\phi^4$ theory, and the second family appear as leading contribution in fishnet field theory introduced by V.Kazakov and O.Gurdogan. The problem of calculation of such type of diagrams can be reformulated as problem of diagonalization of some family of commuting operators. Diagonalization can be performed using methods of quantum integrable systems.