Bäcklund autotransformation for the equation $u_{xt}=e^u-e^{-2u}$

The system of differential equalities arisen in connection with the Bullough–Dodd–Jiber–Shabat equation $u_{xt}=e^u-e^{-2u}$ is considered. It is shown that this system realizes the differential Bäcklund autotransformation for the equation $u_{xt}=e^u-e^{-2u}$. Associated three-dimensional dynamical systems compatible on the two-dimensional invariant submanifold are investigated. Special technique for obtaining their common solutions and the three-parameter soliton of the equation $u_{xt}=e^u-e^{-2u}$ is suggested.