The family of equations
where , , , ,, describes pseudospherical surfaces. It is shown that solutions of (1) correspond to solutions of
Examples include sine-Gordon, sin h- Gordon and Liouville equations. A self-Bäcklund transformation for (2) is constructed based on a geometric method introduced by A. Cavalcante and by L. P. Jorge and the third author [J. Math. Phys. 29, 1044-1049 (1988; Zbl 0695.35038)] and by L. P. Jorge and the third author [Stud. Appl. Math. 77, 103-107 (1987; Zbl 0642.35017)].
Finally, solutions to equation (2), with , and if are obtained using an inverse scattering method.