In the study of Hamiltonian systems, integrable dynamics play a crucial role. Integrability, however, appears to be a delicate property that is not expected to persist under generic small perturbations. Understanding the essence of this fragility presents a compelling task, which turns out to be relevant across different contexts. In this talk, I shall present some results aimed at shedding more light on this issue, within the framework of symplectic twist maps of the 2-dimensional annulus. Specifically, I shall investigate the persistence and the properties of invariant Lagrangian tori that are foliated by periodic points and discuss how their fragility underpins the rigidity of completely integrable twist maps. This is based on a joint works with Marie-Claude Arnaud, Jessica Massetti, and Corentin Fierobe.