与3×3矩阵谱问题相关的有限维可积系统远没有2×2情况的研究成果丰硕，因为对于3×3矩阵谱问题相关的可积系统首先要面对的就是三角曲线，比2×2情形复杂许多。本项目主要讨论与3×3矩阵谱问题相关的有限维Hamilton系统的两个方面：(1) 在Lie-Poisson结构框架下建立的有限维Hamilton系统，与以往在辛结构框架下证明系统的可积性相比该框架简化了可积性的证明。(2) 在Casimir函数的公共水平集上，通过一系列保Poisson 结构的典则变换，利用寻找到的可分离变量，最终结合Hamilton-Jacobi 理论给出作用-角变量及角变量，进而得到Jacobi 反演问题。该课题的探讨将为深入研究高阶谱问题对应可积系统提供新的途径和思路。

The research on finite dimensional integrable system associated with the 3×3 matrix spectral problems is far less than the 2×2 cases. For the integrable systems associated with the 3×3 matrix spectral problems, the first thing to face is the triangular curve. It is much more complicated than 2×2 cases. This project mainly discusses two aspects of the finite-dimensional Hamiltonian systems related to the 3×3 matrix spectrum problems: (1) The finite-dimensional Hamiltonian systems established in framework of the Lie-Poisson structure, this treatment greatly simplify the proves of integrability. (2) On the common level set of Casimir functions, the separable variables are introduced by a series of canonical transformations remain Poisson structure unchanged. Finally, the Hamilton-Jacobi theory are used to give the action-angle variables and the Jacobi inversion problems for soliton equations with angle variables. This idea provides new ways and ideas for in-depth study of integrable systems associated with higher-order spectral problems.