On equivalence of one spin system and two-component Camassa-Holm equation

The work is devoted to studying an equivalence of a two-component Camassa-Holm equation and a spin system being a generalization of Heisenberg ferromagnet equation. It is known that the equivalence between two nonlinear integrable equations provides a possibility of an extended search of their various exact solutions. For Camassa-Holm equation, a method of inverse scattering problem can be applied via a system of linear partial differential equations with scalar coefficients. Contrary to Camassa-Holm equation, the coefficients of linear system corresponding to spin equations are related with symmetric matrix Lax representations. This is why, while establishing an equivalence between two above equations, additional difficulties arise. In view of this, we propose a matrix Lax representation for Camassa-Holm equation in a symmetric space. Employing this result, we establish a gauge equivalence between two-component Camassa-Holm equation and a spin system. We describe a relation between their solutions.