  We introduce two families of soliton hierarchies: the twisted hierarchies
associated to symmetric spaces. The Lax pairs of these two hierarchies are
Laurent polynomials in the spectral variable. Our constructions gives a
hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE),
which is the Gauss-Codazzi equation for n-dimensional submanifolds in Euclidean
(2n-1)-space with constant sectional curvature -1. In fact, the GSGE is the
first order system associated to a twisted Grassmannian system. We also study
symmetries for the GSGE.
