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We propose a system of difference-differential equations related to the self-dual network equation. By utilizing a Bäcklund transformation equation for the Toda lattice, we derive its N -soliton solution under nonvanishing boundary conditions at infinity. We present explicit expressions of four types of its 1-soliton solutions and examine them. Next, we present new formulas for finding rational solutions, and using them, we determine and analyze four types of rational solutions for thesedoi:10.1143/ptp.106.1079 fatcat:fqkwh5737zczfir4dc6ktozz5a