We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painlevé property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems, a special class of quasi-homogeneous systems, which gives its necessary conditions for B-integrability and complete integrability. A consequence is that the nonrational Kovalevskaya exponents imply the nonexistence of Darboux first integrals for two-dimensional natural homogeneous polynomial Hamiltonian systems, which relates the singularity structure to the Darboux theory of integrability.

中文翻译：

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拟齐次微分系统的Kovalevskaya指数，弱Painlevé性质和可积性

我们为准齐次微分系统通过Kovalevskaya指数完全可积分提供了一些必要条件。然后，作为一个应用，我们在多项式微分系统的弱Painlevé性质和代数可积性之间建立了新的联系。此外，我们还针对齐次牛顿系统的Kovalevskaya指数（一类特殊的准齐次系统）制定了更强的定理，为B可积性和完全可积性提供了必要条件。结果是非理性的Kovalevskaya指数暗示了二维自然齐次多项式哈密顿系统不存在Darboux第一积分，这将奇异结构与Darboux可积性理论联系起来。