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

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