  A moving parallel frame method is applied to geometric non-stretching curve
flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable
systems with unitary invariance. These systems consist of a bi-Hamiltonian
modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG)
equation, involving a scalar variable coupled to a complex vector variable. The
Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way
from the beginning in formulating a complex matrix representation of the
tangent space sp(n)/u(n) and its bracket relations within the symmetric Lie
algebra (u(n),sp(n)).
