Darboux transformation and exact solutions for the model of cylindrically symmetrical chiral field

The application of the Darboux transformation method to the integrable model of a cylindrically symmetrical chiral field has been considered. The associated linear system of matrix equations has been proposed and the properties of symmetry for its solution obtained. The necessary form of Darboux transformation has been found and formal one- and N-soliton solutions constructed. With the use of Polhmayer's transformation, a sine-Gordon type equation has been given and an hypothesis about its integrability proposed.