标题： 扩展Clebsch和Manakov模型的对称变量分离 显示英文标题

标题： Symmetric Separation of Variables for the Extended Clebsch and Manakov Models

摘要： 在本文中，使用分离变量（SoV）的双哈密顿理论的向量场方法$Z_i$的修改形式，我们构建了克莱布希模型三维扩展的对称非斯塔克尔分离变量，该模型（在双哈密顿意义上）等价于相互作用的马纳科夫（肖特基-弗拉姆）和欧拉陀螺系统。 对于得到的对称SoV（与之前构建的不对称情况相反），所有的分离曲线都相同，且其亏格为五。 结果表明，对称与不对称情况之间的差异体现在用于构建分离多项式的向量场$Z$的不同形式中。 我们显式地构造了所考虑的对称SoV情况下扩展克莱布希和马纳科夫模型的分离坐标和动量以及阿贝尔型方程。 显示英文摘要

摘要： In the present paper, using a modification of the method of vector fields $Z_i$ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-St\"ackel variable separation for three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields $Z$ used to construct separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.