摘要：

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