  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.
