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

Steady solutions for the Schr\"odinger map equation