  Second order partial differential equations which describe spherical surfaces
(ss) or pseudospherical surfaces (pss) are considered. These equations are
equivalent to the structure equations of a metric with Gaussian curvature $K =
1$ or $K = -1$, respectively, and they can be seen as the compatibility
condition of an associated su(2)-valued or sl(2, R)-valued linear problem, also
referred to as a zero curvature representation. Under certain assumptions we
give a complete and explicit classiffcation of equations of the form $z_{tt} =
A(z, z_x , z_t) z_{xx} + B(z, z_x , z_t ) z_{xt} + C(z, z_x , z_t)$ describing
pss or ss, in terms of some arbitrary differentiable functions. Several
examples of such equations are provided by choosing the arbitrary functions. In
particular, well known equations which describe pseudospherical surfaces, such
as the short pulse and the constant astigmatism equations, as well as their
generalizations and their spherical analogues are included in the paper.
