Integrable system and monodromy

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Heun's differential equation is a standard form of Fuchsian differential equations of second order which have four regular singularities. We studied solutions and monodromy of Heun's differential equation and its generalizations from a viewpoint of integrable system. In particular, we found unexplored solutions of Heun's differential equation by applying middle convolution, which is a transformation of differential equations, and we studied monodromy of the solutions. Moreover we clarified a relationship between Heun's differential equation and the space of initial conditions of the sixth Painleve equation.