研究内容：1非完整系统的刘维尔可积性：将哈密顿-雅可比积分理论从点的邻域推广到辛流形的拉格朗日子流形邻域和非完整系统纤维化约束流形的水平分布邻域，求解该方程方程的完全解，获得系统的作用-角变量，分析系统的可积性；2非完整系统的波哥亚夫林斯基可积性：分析非完整约束对刘维尔可积性的异常影响，引入波哥亚夫林斯基可积性，分类分析非完整系统的可积性和运动方程的可解性；3基于非完整系统对称约化的可积性：利用非完整系统的对称约化与可积性的非对合性，构造约化空间上的可积系统；利用动量映射和双李代数结构，在李代数对偶空间上构造Lax形式运动方程和对合余伴随不变函数；4非完整系统可积性的整体性质和数值积分方法研究：研究约束的非完整性如何影响其单值性，探索非完整系统的单值性与作用-角变量的几何关系。研究意义：突破刘维尔可积性对非完整系统的局限，扩展波哥亚夫林斯基可积性，实现更多非完整系统在更广泛意义下的可积性。

Content：1) Liouville integrability of nonholonomic systems: The Hamilton-Jacobi integral theory is extended from the neighborhood of points to the neighborhood of Lagrangian manifolds of symplectic manifolds and the neighborhood of horizontal distribution of fibred constraint manifolds of nonholonomic systems. By solving the complete solution of the equation, the action-angle variables of the system are obtained and the integrability of the system is analyzed. 2) Bogoyavlenskij integrability of nonholonomic systems: The abnormal influence of nonholonomic constraints on Liouville integrability is analyzed. Bogoyavlenskij integrability is introduced to analyze the integrability of nonholonomic systems and the solvability of equations of motion by type. 3) Integrability analysis based on symmetry reduction for nonholonomic systems: An integrable system in reduced space is constructed by utilizing the non-involution of symmetry reduction and integrability of nonholonomic systems. By means of momentum mapping and double Lie algebraic structure, the Lax form of motion equation and the coadjoint invariant functions in involution are constructed on the dual space of Lie algebra. 4) Global properties and numerical integration method for integrability of nonholonomic systems: The geometric relation between monodromy and action-angle variables of nonholonomic systems is explored by studying how the nonholonomy of constraints affect their monodromy. Significance: To break through the limitation of Liouville integrability on nonholonomic systems, expand Bogoyavlenskij integrability, and realize the integrability of more nonholonomic systems in a broader sense.