Toric Networks, Geometric R -Matrices and Generalized Discrete Toda Lattices

We use the combinatorics of toric networks and the double affine geometric R -matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the R -matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.