  In this paper we use bifurcation methods to construct a new family of
solutions of the binormal flow, also known as the vortex filament equation,
which do not change their form. Our examples are complementary to those
obtained by S. Kida in 1981, and therefore they are also related, thanks to the
so-called Hasimoto transformation, to travelling wave solutions of the 1d cubic
non-linear Schr\"odinger equation.
