Correlation between differential equations, geometric structures, and twistor theory

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An aspect of the twistor theory is to research relations and correspondences between different geometric structures defined by a double fibration. 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 equations associated with cone structures, Clairaut equations, Monge-Ampere equations, equations associated with Lie algebra representations.