We study the integrability of a Hamiltonian system proposed recently by Maciejewski and Przybylska, and which constitutes an extension of one studied (and integrated) by Lagrange. While the previous authors used the differential Galois theory formalism in order to study the integrability of this extended Hamiltonian, we approach the problem from the point of view of singularity analysis. For all values of the parameter (integer in the study of Maciejewski and Przybylska and rational in ours) we are able to show that the system does not possess the Painlevé property, with the exception of the case of the harmonic oscillator. The singularity structure of the Lagrange case is analysed in detail and commented upon.

我们研究了Maciejewski和Przybylska最近提出的哈密顿系统的可积性，它构成了拉格朗日研究（和积分）的一个系统的扩展。尽管先前的作者使用差分伽罗瓦理论形式主义来研究此扩展哈密顿量的可积性，但我们还是从奇点分析的角度来解决这个问题。对于所有参数值（对Maciejewski和Przybylska的研究中为整数，在我们的研究中为有理数），我们能够证明该系统不具有Painlevé性质，但谐波振荡器除外。对拉格朗日案例的奇异结构进行了详细的分析和评论。