In this paper, for the study of integrability, symmetry analysis, group invariant solutions and conservation laws, the Mikhailov–Novikov–Wang equation is considered. Firstly, Painlevé analysis is being employed to study the integrability properties for the considered equation so as to check the possibility that this equation passes the Painlevé test. Secondly, Lie group analysis is studied for finding the symmetries by using Lie classical group analysis method and to obtain its symmetry group, infinitesimal generator, Lie algebra commutation table, and similarity reductions. The vector fields and the symmetry reduction of this equation are calculated with the aid of Lie symmetry analysis. From the similarity reduction equation, some explicit exact solutions are derived. Finally, using the new conservation theorem proposed by Ibragimov [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007) 311–328], the conservation laws of the aforesaid equation have been constructed.

中文翻译：

---

Painlevé 分析、群不变分析、相似性约简、精确解和 Mikhailov-Novikov-Wang 方程的守恒定律

在本文中，为了研究可积性、对称性分析、群不变解和守恒定律，考虑了 Mikhailov-Novikov-Wang 方程。首先，Painlevé 分析用于研究所考虑方程的可积性，以检查该方程通过 Painlevé 检验的可能性。其次，研究李群分析，利用李经典群分析方法寻找对称性，得到其对称群、无穷小生成元、李代数交换表和相似性约简。借助李对称分析计算该方程的矢量场和对称约简。从相似性降低方程中，导出了一些明确的精确解。最后，利用伊布拉吉莫夫提出的新守恒定理 [NHJ.数学。肛门。应用程序。 333(2007) 311-328]，已经构建了上述方程的守恒定律。