标题： 双辛对称链的余秩一和相关的Liouville可积系统 显示英文标题

标题： Bi-presymplectic chains of co-rank one and related Liouville integrable systems

摘要： 考虑了一阶形式的余秩为一的双辛链。 推导了这类链表示某些Liouville可积系统的情况以及存在相关向量场的双哈密顿链的情况。 为了展示双辛链的构造，使用了对偶泊松-辛对的概念，并引入了泊松二向量的d相容性和辛形式的d相容性的概念。 表明相关流的双辛表示可以直接以纯算法方式构造分离坐标。 作为示例，详细考虑了${\mathbb R}^3$中的双辛和双哈密顿链。 显示英文摘要

摘要： Bi-presymplectic chains of one-forms of co-rank one are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of dual Poisson-presymplectic pair is used and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of related flow leads directly to the construction of separation coordinates in purely algorithmic way. As an illustration bi-presymplectic and bi-Hamiltonian chains in ${\mathbb R}^3$ are considered in detail.