Abstract In this paper, we study complex isochronous center problem for cubic complex planar vector fields, which are assumed to be Z 2 -equivariant with two symmetric centers. Such integrable systems can be classified as 11 cases. A complete classification is given on the complex isochronous centers and proven to have a total of 54 cases. All the algebraic conditions for the 54 cases are derived and, moreover, all the corresponding linearization transformations are obtained. This problem for the Z 2 -equivariant with two symmetric centers has been completely solved.

中文翻译：

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三次 Z2 等变平面系统的复等时中心和线性化变换

摘要 在本文中，我们研究了三次复平面向量场的复等时中心问题，假设它们是具有两个对称中心的Z 2 -等变的。这种可积系统可分为11种情况。对复杂的等时中心给出了完整的分类，并证明总共有54个案例。推导出了54种情况的所有代数条件，并获得了所有相应的线性化变换。具有两个对称中心的 Z 2 -等变体的这个问题已经完全解决。