A biperiodic planar resistor network is a pair (G, c) where G is a graph embedded on the torus and c is a function from the edges of G to non-zero complex numbers. Associated with the discrete Laplacian on a biperiodic planar network is its spectral data: a triple \((C,S,\nu )\) , where C is a curve, and S is a divisor on it, which we show is a point in the Prym variety of C. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of cluster automorphisms arising from the Y- \(\Delta \) transformation, giving discrete cluster integrable systems. We show that these automorphisms are integrable in the algebro-geometric sense: under the spectral transform, they become translations in the Prym variety.
