众所周知，“可积性”是微分方程研究领域中一个基本而又重要的问题。近几十年来，由于混沌的发现促使人们提高了对不可积性系统的研究兴趣。沿着 Poincaré的思路展开了很多用来判断首次积分不存在的准则，但这些工作大都是在系统不满足共振条件或者满足某种特殊共振条件下展开的。另外，物理学中的很多动力系统有一个或两个首次积分，则其不可积性一般是很难证明的，因此研究系统的部分可积性有着重要的意义。在本项目中我们将以Poincaré法型理论为主要工具，结合Ziglin引理、Floquent理论以及半拟齐次系统的相关结果来研究微分方程系统满足一般共振条件的有理首次积分的不存在性和部分存在性。进一步对拟周期系统在满足共振情形下给出有理首次积分相关判定准则。另外，关于不可积性已有结果大多是在奇点附近做出的研究，我们将在周期解附近给出微分方程系统存在有理首次积分的必要条件和系统部分可积的判定准则。

As is well known, “integrability” is a fundamental and significant topic in the field of studying differential equation. In recent decades, due to the discovery of chaos, nonintegrable systems have been received much attention by researchers. Along the idea of Poincaré, a lot of works have been expanded. But most of works all considered the cases that the systems are not resonant or simply resonant. On the other hand, many dynamical systems in physics have one or two first integrals, its nonintegrability in general is difficult to be proved, so the research of partial integrability has a very important significance. In this project, we will use the theory of Poincaré normal form as a main tool, combined with the Ziglin lemma, Floquent theory and related results of semi-quasi-homogeneous system, to study the nonexistence and partial existence of rational first integrals for differential equation systems under the general resonance condition. Furthermore, we will also give the correlative criteria of rational first integrals for quasi-periodic systems under the case of resonance. In addition, since the existence results about the nonintegrability system are mostly considered near the singular point, we will give the criteria about the nonexistence and partial existence of rational first integrals near the periodic solution.