Analysis of an algebraic structure of a degenerate Garnier system from a viewpoint of algebraic solutions

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(1) We consider degenerate Garnier systems G(3,1,1) (which is defined by H.Kimura) and G(5/2,1,1) (which is defined by H.Kawamuko), and find all rational solutions of each equation.(2) We consider degenerate Garnier system G(3,2) (which is defined by H.Kimura), and find all algebraic solutions of G (3,2).(3) By using the results (1) and (2), we show that there is no birational transformation between these systems.(4) We consider a third order Fuchsian differential equation which has three regular singular points on Riemann sphere, and show that the Schlesinger transformation is equivalent to a difference VI Painlev\'e equation. Moreover, we can define a difference Garnier system in a similar way.