Eliasson's theorems on normal forms at nondegenerate singularities are central to the study of (smooth) integrable systems. The absence of a unified treatment of the proof has led us to explore different alternatives: a geometric approach using the Moser path method for integrable slightly differently than in the literature and a variant of the Nash-Moser fast convergence method. The latter is formulated as a meta theorem about rigidity, involving closed pseudogroups and PDEs.
Several examples of the rigidity theorem are presented (in order to demonstrate the theorem and its connection with normal form theorems), including a new proof of the Newlander-Nirenberg theorem. Some complications prevented us from applying this theorem to integrable systems, which we hope to overcome for the article version of this work.