In the present paper we develop a method of the vector fields $Z_i$ [9] in the theory of separation of variables. For an integrable case of the complex Kirchhoff's problem on $e^{⁎}(3)$, which has been never considered before, we construct—with the help of this method—two types of separation of variables (SoV): symmetric and asymmetric ones. Our asymmetric SoV is unusual: it is characterized by the quadratures containing differentials defined on two different curves of separation. It is a direct analogue of asymmetric SoV for the Clebsch model [17]. In the case of symmetric SoV both curves of separation are the same. This case has an additional bonus: on zero level set of one of the Casimir functions it yields a direct analogue of the famous Weber-Neumann separated coordinates. They are also considered in the present paper in some details.

在本文中，我们开发了一种矢量场的方法 $Z_{\mathrm{一世}}$[9] 在变量分离理论中。对于复基尔霍夫问题的可积情况${\mathrm{电子}}^{⁎}(3)$，这是以前从未考虑过的，我们在这种方法的帮助下构建了两种类型的变量分离 (SoV)：对称和非对称。我们的非对称 SoV 是不寻常的：它的特点是包含定义在两条不同分离曲线上的差分的正交。它是 Clebsch 模型 [17] 的非对称 SoV 的直接模拟。在对称 SoV 的情况下，两条分离曲线是相同的。这种情况有一个额外的好处：在一个 Casimir 函数的零水平集上，它产生了著名的 Weber-Neumann 分离坐标的直接模拟。在本文中也详细讨论了它们。