We study the higher KdV flows on the space of closed complex equicentroaffine curves as Hamiltonian systems. Using a suitable presymplectic structure of the space, we give the Hamiltonian flows associated with the higher KdV equations and a map between the space and the space of closed curves in the complex plane, which induces the Miura transformation between the higher KdV equations and the higher mKdV equations.
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