We prove that for any autonomous 4-dimensional integral system of Painleve type, the Jacobian of the generic spectral curve has a unique polarization, and thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian surface to any other Jacobian. This enables us to identify the spectral curve and any irreducible genus two component of the boundary of an affine patch of the Liouville torus.

中文翻译：

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自主 4 维 Painlevé 型系统偏振的唯一性

我们证明，对于任何 Painleve 类型的自治 4 维积分系统，通用谱曲线的雅可比矩阵具有唯一的极化，因此根据托雷利定理，不能同构为任何其他雅可比矩阵的非极化阿贝尔曲面。这使我们能够识别 Liouville 环面仿射斑块边界的光谱曲线和任何不可约的第二属成分。