In this paper, we work out the reciprocal transformations generated by the adjoint eigenfunctions, and write down the corresponding changes of the Lax operators, eigenfunctions and adjoint eigenfunctions under this kind of new reciprocal transformations at first. Then we discuss the successive actions of the reciprocal transformations generated by the eigenfunctions and the adjoint eigenfunctions. Also we investigate the commutativity of different types of the reciprocal transformations. The above results are very useful to construct the new type solutions.

中文翻译：

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哈里-戴姆等级制的对等变换

在本文中，我们计算了伴随本征函数产生的倒数变换，并首先写下了在这种新的倒数变换下Lax算子，本征函数和伴随本征函数的相应变化。然后，我们讨论由本征函数和伴随本征函数生成的倒数变换的相继作用。我们还研究了互逆变换的不同类型的可交换性。以上结果对于构造新型解决方案非常有用。