Studies on Painleve or Garnier systems by means of their phase spaces

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The purpose of this assignment is to clarify the fundamental properties of Painleve or Garnier systems by means of their phase spaces. Garnier systems of two independent variables are defined corresponding to partitions of 5, and there exists confluence process between any two systems for the partitions one of which is adjacent to the other. The most important result of our study is to lift up the confluence processes to the level of phase spaces for Garnier systems of two variables. We have tried to prove the Painleve property for the degenerate Garnier systems of two variables by using the above result, but complete proof has not yet been obtained.