Summary: Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the form \[ \nu z_t - \lambda z_{x x t} = A(z, z_x, z_{x x}) z_{x x x} + B(z, z_x, z_{x x}), \] with \(\nu, \lambda \in \mathbb{R}\), \(\nu^2 + \lambda^2 \neq 0\) and \(A^2 + B^2 \neq 0\), 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. In particular, the problem of determining sequences of conservation laws, either in the ss or pss case, is discussed and illustrated by some examples.