  In this paper we consider surfaces with one or two families of spherical
curvature lines. We show that every surface with a family of spherical
curvature lines can locally be generated by a pair of initial data: a suitable
curve of Lie sphere transformations and a spherical Legendre curve. We then
provide conditions on the initial data for which such a surface is Lie
applicable, an integrable class of surfaces that includes cmc and
pseudospherical surfaces. In particular we show that a Lie applicable surface
with exactly one family of spherical curvature lines must be generated by the
lift of a constrained elastic curve in some space form. In view of this goal,
we give a Lie sphere geometric characterisation of constrained elastic curves
via polynomial conserved quantities of a certain family of connections.
