$1+1$ Gaudin Model

We study $1+1$ field-generalizations of the rational and elliptic Gaudin models. For ${\rm sl}(N)$ case we introduce equations of motion and L-A pair with spectral parameter on the Riemann sphere and elliptic curve. In ${\rm sl}(2)$ case we study the equations in detail and find the corresponding Hamiltonian densities. The $n$-site model describes $n$ interacting Landau–Lifshitz models of magnets. The interaction depends on position of the sites (marked points on the curve). We also analyze the $2$-site case in its own right and describe its relation to the principal chiral model. We emphasize that $1+1$ version impose a restriction on a choice of flows on the level of the corresponding $0+1$ classical mechanics.