We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

我们提供某些类别的不连续分段微分系统的最大极限环数，该系统由两个以直线分隔的微分系统组成，当这些微分系统是线性中心或三族立方等时中心时，会产生十种不同类别的不连续分段微分系统。极限循环的最大数量取决于选择的类别，范围从0、1、2、3、5、7和12。对于其中的九种，我们证明达到了相应的最大极限循环数。尤其是，我们已经解决了将第16希尔伯特问题的第二部分扩展到这些类别的不连续分段微分系统。