Defining manifolds and symmetries of nonlinear special functions in several variables

1.构造了Garnier系统及其所有退化系统的二元定义流形。我们还解决了A^<（1）>_4型Noumi-Yamada系统的同样问题。2.我们证明了每一个Painleve方程的Backlund变换群所定义的流形与K. Okamoto所构造的相应定义流形同构。3.我们证明了Painleve方程的Backlund变换群中存在一个层次，即：所有Painleve方程的Backlund变换群都可以从第六类Painleve方程的Backlund变换群出发，利用通常的汇合方法依次得到. 4.我们利用Riemann-Hilbert对应刻画了第六类Painleve方程的Backlund变换群，即，第六类Painleve方程的相空间（初始条件空间）到monodromy表示的模空间的对应，就是以D^（1）>_4型仿射Weyl群为覆盖变换群的覆盖映射。

1.We have constructed the defining manifolds of the Garnier system and all its degenerate systems in two varibales. We have also solved the same problem for the Noumi-Yamada system of type A^<(1)>_4.2.We have proved that the manifold defined by means of Backlund transformation group for each Painleve equation is isomorphic to the corresponding defining manifold constructed by K.Okamoto.3.We have shown that there exists an hierarchy in the Backlund transformation groups for Painleve equations, namely, the Backlund transformation groups for all Painleve equations can be obtained successively from that for the sixth Painleve equation by the use of the usual confluence procedures.4.We have characterized the Backlund transformation group for the sixth Painleve equation by means of Riemann-Hilbert correspondence, namely, the correspondence from the phase space(the space of initial conditions) of the sixth Painleve equation to the moduli space of the monodromy representation is just a covering mapping with the affine Weyl group of type D^<(1)>_4 as the covering transformation group.