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A moving parallel frame method is applied to geometric non-stretching curveflows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrablesystems with unitary invariance. These systems consist of a bi-Hamiltonianmodified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG)equation, involving a scalar variable coupled to a complex vector variable. TheHermitian structure of the symmetric space Sp(n)/U(n) is used in a natural wayfrom the beginning in formulating a complex matrix representation of thetangent space sp(n)/u(n) and its bracket relations within the symmetric Liealgebra (u(n),sp(n)).

Integrable Systems with Unitary Invariance from Non-stretching Geometric Curve Flows in the Hermitian Symmetric Space Sp(n)/U(n)