Nonlinear $\sigma$-model in a curved space, gauge equivalence, and exact solutions of $(2+0)$-dimensional integrable equations

We propose a nonlinear $\sigma$-model in a curved space as a general integrable elliptic model. We construct its exact solutions and obtain energy estimates near the critical point. We consider the Pohlmeyer transformation in Euclidean space and investigate the gauge equivalence conditions for a broad class of elliptic equations. We develop the inverse scattering transform method for the $\operatorname {sh}$-Gordon equation and evaluate its exact and asymptotic solutions.