  In this paper, generalizing the techniques of Bour's theorem, we prove that
every generic cuspidal edge, more generally, generic $n$-type edge, which is
invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial
isometric deformations. As a corollary, several geometric invariants, such as
the limiting normal curvature, the cusp-directional torsion, the higher order
cuspidal curvature and the bias, are proved to be extrinsic invariants.
