In this talk, invariant curve flows in centro-affine symplectic geometry and symplectic Grassmannian space are studied. Symplectic invariants for curves in those geometries are derived by developing the moving frame method. It is shown that several integrable systems arise naturally from the intrinsic curve flows in centro-affine symplectic geometry and symplectic Grassmannian space. The integrability and solutions to those integrable systems are also discussed.

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