We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how they often give rise to regular Fr��licher Lie groups. Then, we show how such deformed algebras arise in a formal integration of Lax equations through time scaling. The infinite dimensional groups under consideration enables to state the uniqueness of the formal solutions, their smooth dependance under perturbation, and to study some of the symmetries.

extension of some results of a published paper available at arXiv:1007.3543; the main result of arXiv:1101.2370 is added in appendix