  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.
