  A moving frame formulation of geometric non-stretching flows of curves in the
Riemannian symmetric spaces $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ is
used to derive two bi-Hamiltonian hierarchies of symplectically-invariant
soliton equations. As main results, multi-component versions of the sine-Gordon
(SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting
$Sp(1)\times Sp(n-1)$ invariance are obtained along with their bi-Hamiltonian
integrability structure consisting of a shared hierarchy of symmetries and
conservation laws generated by a hereditary recursion operator. The
corresponding geometric curve flows in $Sp(n+1)/Sp(1)\times Sp(n)$ and
$SU(2n)/Sp(n)$ are shown to be described by a non-stretching wave map and a
mKdV analog of a non-stretching Schr\"odinger map.
