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Primitive harmonic maps of finite type from a Riemann surface M into a k-symmetric space G/H are obtained by integrating a pair of commuting Hamiltonian vector fields on certain finite-dimensional subspaces of loop algebras. We will clarify and generalize Ohnita and Udagawa's results concerning homogeneous projections p:G/H→G/K, with H⊂K, preserving finite-type property for primitive harmonic maps

Twistor fibrations giving primitive harmonic maps of finite type