Abstract

It is shown that Liouville foliations of the family of non-Euclidean analogs of Kovalevskaya integrable system on a pencil of Lie algebras have both compact and noncompact fibers. There exists a bifurcation of their compact common level surface into a noncompact one that has a noncompact singular fiber. In particular, this is true for the non-Euclidean \(e(2,1)\)-analog of the Kovalevskaya case of rigid body dynamics. In the case of nonzero area integral, an effective criterion for existence of a noncompact connected component of the common level surface of first integrals and Casimir functions is proved.

中文翻译：

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李代数铅笔的纤维非紧性和非欧氏Kovalevskaya系统的奇点

摘要

结果表明，Lie代数铅笔上的Kovalevskaya可积系统的非欧几里得类似物家族的Liouville叶片既具有致密纤维，又具有非致密纤维。它们的紧凑的公共水平面分叉成具有非紧凑的奇异纤维的非紧凑的水平面。特别是对于刚体动力学的Kovalevskaya情况的非欧几里德\（e（2,1）\）-模拟是正确的。在非零面积积分的情况下，证明了存在第一积分和卡西米尔函数的公共水平面的非紧连通分量的有效判据。