Twistor correspondence between different geometric structures and application to differential equations and field theory

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We regard the twister theory as the correspondence via a double fibering, which is the duality between geometric structures on different spaces. From this point of view, we applied to the study of the essence and the construction of equations themselves and solutions for various differential equations and integrable field theory.1.Extending Monge-Ampere equations which are Hessian=const. and Gaussian curvature=const., we intrinsically defined and studied Monge-Ampere systems with Lagrangian pair on contact manifolds. In the case of 5, 7 dimensins, we saw that generic geometric solutions have four and eleven kinds of singularities respevtively via Legendre duality.2.A Goursat equation is a second order PDE whichi is of parabolic type and the Monge characteristic of which is completely integrable. It has the interpretation of twistor theory. We constructed equations themselves by Lagrange-Grassmann duality and solutions by Cartan-Legendre duality.3.We saw that Clairaut equations are nothing but the essence of twistor theory. We considered the various extention by the method of twistor theory.4.Nondegenerate type type (4,7) distributions are of finite type unlike contact structures. We constructed the normal Cartan connections and they have two curvature invariants. Hyperbolic type distributions have relation to Legendre geodesies via twistor theory.5.We define SU(3) type U(1) instantons on 6-dimensional manifolds with SU(3) structure.In particular, in the case of nearly Kahler structure, we constructed SU(3) type U(1) instantons on Hopf bundles of CP^3 and F_{12} which are the twistor space of S^4 and CP^2 respectively.