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It is shown that the kinematic system describing planar non-steady motions ofideal fibre-reinforced fluids may be reduced to a single two-dimensionalthird-order partial differential equation in which time enters parametrically.A procedure is established which maps steady motions to non-steady motions. Theresulting motions inherit their hidden integrable structure from the steadycase. The formalism presented here also readily recovers the connection withthe scattering problem of the modified Korteweg-de Vries hierarchy establishedin previous work.

On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure