摘要：

In this report, we introduce a method for directly constructing Birkhoffsections on energy surfaces of integrable Hamiltonian systems based on theirdynamical behavior. We establish the necessary and sufficient conditions for varioustypes of periodic orbits to serve as boundary orbits of a Birkhoff section. Thisconstruction applies to the boundaries of toric domains and energy surfaces ofseparable Hamiltonian systems, providing criteria for the existence or non-existenceof different types of Birkhoff sections.