An evolution equation is said to describe pseudospherical surfaces if it is a necessary and sufficient condition for the existence of functions , ; , depending on , , and its derivatives, such that one-forms satisfy the structure equations of a surface of constant Gaussian curvature equal to , with metric and connection one-form . On the other hand, an equation is said to be formally integrable if it has a formal symmetry of infinite rank.
The author shows that every second-order equation which is formally integrable, describes a one-parameter family of pseudospherical surfaces. To this end, he finds explicitly the pseudospherical structures associated with each of the four second-order equations appearing in the exhaustive list of formally integrable equations available in literature. It is shown that this result cannot be extended to third-order formally integrable equations. This fact notwithstanding, a special case of the equation of the form
is considered, and every formally integrable equation of this form is proved to describe a one-parameter family of pseudospherical surfaces. Finally, several popular nonlinear equations, including the Harry-Dym, cylindrical KdV and the Calogero-Degasperis family are shown to describe pseudospherical surfaces.