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

Toric networks, geometric $R$-matrices and generalized discrete Toda lattices