On Integrable Systems & Rigidity for PDEs with Symmetry

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We motivate the definition of weak equivalence (found in the literature) by studying different interpretations of the concept `singular Lagrangian foliation'. Finally, we prove a normal form theorem inspired by the Moser path method. In the second part we study the deformation theory of a PDE with a pseudogroup of symmetries in general. We prove that a solution is rigid if its deformation cohomology vanishes and certain `tame' estimates hold, inspired by the Nash-Moser fast convergence method. An essential detail is that this statements holds for local solutions of the PDE around a compact submanifold. We give alternative proofs for several classical normal form theorems, as applications. Key Words: completely integrable systems, singular foliations, singular Lagrangian foliations, closed pseudogroups, Lie algebra sheaves, geometric PDEs, local rigidity, normal forms, Nash-Moser, Moser path method, Newlander-Nirenberg, Eliasson.