Construction of pluriharmonic maps from complex torus into symmetric spaces and applications of the theory of integrable systems

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In our study, I obtained three main results. I expain each of them separately.(1) F. Burstall proved that any weakly conformal non-isotropic harmonic map of 2-torus into a sphere or a complex projective space can be lifted to a primitive map of finite type into some k-symmetric space. We generalized the result of F. Burstall to obtain that any weakly conformal non-isotropic harmonic map of 2-torus into a sphere or a complex projective space is itself of finite type. In fact, we can prove a more general result. Given a primitive map of finite type into a generalized flag manifold, we can project it into some compact symmetric space as a harmonic map of finite type under some condition on the choice of isotropy subgroup of the compact symmetric space. The condition is rather mild and satisfied by the above cases (except odd-dimensional sphere. But, this case is included the case of even-dimensional sphere).(2) We extended the result (1) to the case where the domain is a complex manifold and pluriharmonic maps into it. We proved that any non-isotropic pluriharmonic map of complex torus into a complex projective space is of finite type.(3) We introduced the concept of primitive maps of generalized finite type and obtained some results on the harmonic maps of compact Riemann surfaces of higher genus. In fact, any primitive harmonic maps of generalized finite type of a compact Riemann surface of genus greater than 1 into a k-symmetric space is a composition of a primitive pluriharmonic map of finite type of some Jacobian torus into a k-symmetric space with a Abel map of the compact Riemann surface into the Jacobian torus.