Summary: The purely binormal motion of curves of constant curvature or torsion, respectively, is shown to lead to integrable extensions of the Dym and classical sine-Gordon equations. In the case of the extended Dym equation, a reciprocal invariance is used to establish the existence of novel dual-soliton surfaces associated with a given soliton surface. A cc-ideal formulation is adduced to obtain a matrix Darboux invariance for the extended Dym and reciprocally linked \(m^2\)KdV equations. A Bäcklund transformation with a classical-length property is thereby constructed, which allows the generation of associated soliton surfaces. Analogues of both Bäcklund’s and Bianchi’s classical transformations are derived for the extended sine-Gordon system.