Bonnet surfaces are those surfaces in Euclidean 3-space \({\mathbf E}^3\) that admit at least one nontrivial mean curvature preserving isometry \(\Phi\) (here ``nontrivial'' means that \(\Phi\) can not be extended to an isometry of \({\mathbf E}^3\)). A special class \((C3)\) among these surfaces are the ones that admit exactly one such isometry. A general criterion for a surface to be within \(C3\) is given. With the help of this criterion tangential developable surfaces belonging to \(C3\) can be implicitly characterized by elliptic integrals of the third kind. Things get simpler in the following example: if \(S\) is a tangential developable surface with a screw line as cuspidal edge then \(\Phi\) can be given explicitly and moreover \(\Phi(S) = S\).