Study on relations between differential equations and geometric structures by the method of twistor theory

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An observation of twistor theory is to research relations and correspondences between different geometric structures via double fibrations. For various classes of differential equations associated with geometric structures, we study geometric meanings of equations, properties of solutions, constructions of equations and solutions. We discuss differential equations associated with cone structures, Monge-Ampere systems, equations associated with Lie algebra representations, equations associated with integrals of powers of polynomials. Treating with conformal triality from D_4 twistor diagram, we study the geometric structures, singularities, and differential equations. Treating with Kaluza-Klein space-time with (3,3) type, we study the scalar field, the vector field, the spinor field induced by the conformal SO(4,4) representation.