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We present a steady Euler flow on the round 3-sphere whose velocity vectorfield has the property of having two independent first integrals, being tangentto the fibres of an almost submersion onto the 2-sphere. This submersion turnsout to be a critical point for the quartic Faddeev-Skyrme model with a standardpotential.

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution