Quasi-Homogeneous Integrable Systems: Free Parameters, Kovalevskaya Exponents, and the Painlev\'e Property

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.