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Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in 
${{\mathbb{R}}^{3}}$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in 
${{H}^{3}}$ or 
${{S}^{3}}$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.
