  Third order equations, which describe spherical surfaces (ss) or
pseudospherical surfaces (pss), of the form \[
\nu\,z_{t}-\lambda\,z_{xxt}=A(z,z_{x},z_{xx})\,z_{xxx}+B(z,z_{x},z_{xx}) \]
with $\nu$, $\lambda$ $\in$ $\mathbb{R}$, $\nu^2+\lambda^2\neq 0$, are
considered. These equations are equivalent to the structure equations of a
metric with Gaussian curvature $K=1$ or $K=-1$, respectively. Alternatively
they can be seen as the compatibility condition of an associated
$\mathfrak{su}(2)$-valued or $\mathfrak{sl}(2,\mathbb{R})$-valued linear
problem, also referred to as a zero curvature representation. Under certain
assumptions we obtain an explicit classification for equations of the
considered form that describe ss or pss, in terms of some arbitrary
differentiable functions. Several examples of such equations, which describe
also a number of already known equations, are provided by suitably choosing the
arbitrary functions.
