标题： 拟齐次可积系统：自由参数、Kovalevskaya 指数和Painlevé性质 显示英文标题

标题： Quasi-Homogeneous Integrable Systems: Free Parameters, Kovalevskaya Exponents, and the Painlevé Property

摘要： 本文通过分析四维Painlevé型方程的Kovalevskaya指数所观察到的模式，研究了拟齐次可积系统在其可移动奇点附近的Laurent级数解。 我们引入了一个参数空间来编码这些展开中的自由系数，并研究其在交换拟齐次向量场下的变形。 在此框架下，我们从主指标簇导出较低阶的指标簇，并建立了一种关于Kovalevskaya指数的算术共振条件，该条件支配着分数幂的出现以及Painlevé性质的失效。 此外，我们通过初值映射在参数空间上构造了一个Frobenius流形结构，当所有权值相同时，该结构变为共形的。 在Hamiltonian背景下，我们证明了参数空间上的诱导流保持辛形式，并自然地给出了Kovalevskaya指数的配对。 这些发现统一了拟齐次可积系统的解析和几何方面，并为理解其形变理论和奇点结构提供了新的见解。 我们的结果为Painlevé型方程及相关可积模型的分类与分析提供了一个全面的框架。 显示英文摘要

摘要： This paper investigates quasi-homogeneous integrable systems by analyzing their Laurent series solutions near movable singularities, motivated by patterns observed in Kovalevskaya exponents of four-dimensional Painlev\'e-type equations. We introduce a parameter space encoding the free coefficients in these expansions and study its deformation under a commuting quasi-homogeneous vector field. Within this framework, we derive lower indicial loci from the principal one and establish an arithmetic resonance condition on Kovalevskaya exponents that governs the emergence of fractional powers and the breakdown of the Painlev\'e property. Moreover, we construct a Frobenius manifold structure on the parameter space via the initial value map, which becomes conformal when all weights coincide. In the Hamiltonian context, we demonstrate that the induced flow on the parameter space preserves a symplectic form and yields a natural pairing of Kovalevskaya exponents. These findings unify analytic and geometric aspects of quasi-homogeneous integrable systems and offer new insights into their deformation theory and singularity structures. Our results provide a comprehensive framework applicable to the classification and analysis of Painlev\'e-type equations and related integrable models.