本项目研究包含二维仿射Toda方程的具有复杂对称性的可积系统的Darboux变换以及近年来受到广泛关注的非局部可积系统的整体精确解的系统构造。主要内容为：(1) 对目前尚不清楚其Darboux变换构造的相应于两族仿射Lie代数 A_{2n-1}^{(2)}和 D_n^｛(1)｝的二维Toda方程，完成保持其Lax对的所有对称性的Darboux变换的构造，并研究具有类似对称性的可积系统的Darboux变换。(2) 对1+1维和2+1维非局部可积系统，研究如何找到Darboux变换的参数所满足的条件，从而能不通过具体选择参数而由任意次Darboux变换系统地构造整体精确解，进而研究当时间趋于无穷时经任意次Darboux变换所得精确解的渐近性质，同时，通过建立高维非局部可积系统与低维非局部可积系统的联系，由低维系统的整体精确解来求得高维系统的整体精确解。

In this project we will study the Darboux transformations of integrable systems with complicated symmetries which consist of the two dimensional affine Toda equations, and the systematic construction of global exact solutions of nonlocal integrable systems. The main subjects are as follows. (1) Construct the Darboux transformations keeping all the symmetries for the two dimensional Toda equations corresponding to two series of affine Lie algebras A_{2n-1}^{(2)} and D_n^｛(1)} and related integrable systems with similar symmetries. (2) For 1+1 and 2+1 dimensional nonlocal integrable systems, find the conditions for the parameters of Darboux transformations so that the exact solutions given by Darboux transformations of arbitrary order are all globally defined. Then study the asymptotic behavior of the derived global exact solutions as time tends to infinity. By establishing the relation between nonlocal integrable systems of higher dimension and of lower dimension, construct the global exact solutions of higher dimensional nonlocal integrable system from those of lower one.