本项目将从无穷阶矩阵形式的bigraded Toda系统约化到有限带状矩阵形式的Toda系统。 我们将从其正则tau函数的表示形式考虑其渐近行为，也就是给出有限带状矩阵解的Sorting 性质。. 我们还将考虑有限带状矩阵的谱问题对应的Partial旗流形刻化。众所周知，Full Kostant-Toda 系统对应着标准旗流形，Hessenberg形式的非满矩阵对应着Partial旗流形。对于一般的带状结构矩阵，对应着什么样的旗流形以及这样的旗流形的Bruhat分解将是我们本项目的研究内容之一。. 我们还将构造从有限阶bigraded Toda系统的tau函数到多面体的矩映射，在多面体上分析有限阶bigraded Toda系统的各种流的走向以及奇点等。

In this project, we will do a reduction from the bigraded Toda system in infinite-sized matrices to the Toda system in finite-sized Band matrices. We will further study the asymptotic behavior of the solution in terms of regular tau functions, i.e. the sorting property of solutions in finite-sized Band matrices.. We will further consider the Partial flag manifolds which correspond to the Toda system in finite-sized Band matrices. It is well-known that full Kostant-Toda system corresponds to the standard flag manifold, the Toda system in nonfull Hessenberg forms corresponds to the partial flag manifold. For the general Band matrices, what is the corresponding flag manifold and how to do the Schubert decomposition of the flag variety will be included in our project.. After this, we will also construct the moment map from finite-sized Band Toda system to polytopes and do analysis on the Toda flows and singularities.