We study the integrability of an eight-parameter family of three-dimensional spherically confined steady Stokes flows introduced by Bajer and Moffatt. This volume-preserving flow was constructed to model the stretch–twist–fold mechanism of the fast dynamo magnetohydrodynamical model. In particular we obtain a complete classification of cases when the system admits an additional Darboux polynomial of degree one. All but one such case are integrable, and first integrals are presented in the paper. The case when the system admits an additional Darboux polynomial of degree one but is not evidently integrable is investigated by methods of differential Galois theory. It is proved that the four-parameter family contained in this case is not integrable in the Jacobi sense, i.e. it does not admit a meromorphic first integral. Moreover, we investigate the integrability of other four-parameter \({\textit{STF}}\) systems using the same methods. We distinguish all the cases when the system satisfies necessary conditions for integrability obtained from an analysis of the differential Galois group of variational equations.

中文翻译：

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拉伸-扭曲-折叠流动的可积性分析

我们研究了由Bajer和Moffatt引入的八参数族的三维球形受限稳态Stokes流的可积性。构造该保留体积的流是为了模拟快速发电机磁流体动力学模型的拉伸-扭曲-折叠机制。特别是当系统接受一个附加的一阶Darboux多项式时，我们获得了一个完整的案例分类。除一个这样的情况外，所有其他情况都是可积的，并且在本文中介绍了第一积分。通过微分伽罗瓦理论的方法研究了系统允许一个额外的度数的Darboux多项式但显然不可积的情况。证明在这种情况下包含的四参数族在雅可比意义上是不可积分的，即，它不容许亚纯的第一积分。此外，\（{\ textit {STF}} \）系统使用相同的方法。当系统满足通过对微分Galois变分方程组的分析而获得的可积性的必要条件时，我们将区分所有情况。