Abstract For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described.

中文翻译：

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两步自由幂零李群的周期时间最优控制

摘要 对于两步自由幂零李代数，我们描述了辛叶和 Casimir 函数。考虑一个左不变的时间最优问题，其中允许控制的集合由李代数第一层中的严格凸紧集给出，该集合包含其内部的原点。我们描述了庞特里亚金最大值原理的哈密顿系统的垂直子系统的积分。描述了该系统对于低阶泊松双向量的解的性质。