We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N_u,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group S_n and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for N_u,v represent elements of the Coxeter double Bruhat cell G^u,v⊂GL_n. The Cartan subgroup H acts on G^u,v by conjugation. The standard Poisson structure on the space of weights of N_u,v induces a Poisson structure on G^u,v, and hence on the quotient G^u,v/H, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for N_u,v. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.

Following the ideas developed in our previous papers, we introduce a cluster algebra $ \mathcal{A} $ on $ {\mathcal{R}_n} $ compatible with the obtained Poisson bracket. Generalized Bäcklund–Darboux transformations map solutions of one Coxeter–Toda lattice to solutions of another preserving the corresponding Weyl function. Using network representation, we construct generalized Bäcklund–Darboux transformations as appropriate sequences of cluster transformations in $ \mathcal{A} $.