Lagrangian fibers of Gelfand--Cetlin systems of $\mathrm{S}\mathrm{O}(n)$-type

In this paper, we study the Gelfand–Cetlin systems and polytopes of the co-adjoint $\mathrm{S}\mathrm{O}(n)$-orbits. We describe the face structure of Gelfand–Cetlin polytopes and iterated bundle structure of Gelfand–Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.