Local Integrable Symmetries of Diffieties

In the framework of diffieties, introduced by Vinogradov, we introduce integrable infinitesimal symmetries and show that they define a one parameter pseudogroup of local diffiety morphisms. We prove some preliminary results allowing to reduce the computation of integrable infinitesimal symmetries of a given order to solving a system of partial differential equations.We provide examples for which we can reduce to a linear system that can be solved by hand computation, and investigate some consequences for the local classification of diffiety, with a special interest for testing if a diffiety is flat.

💡 Research Summary

The paper develops a novel theory of “integrable infinitesimal symmetries” within the geometric framework of diffieties, originally introduced by Vinogradov. A diffiety is an infinite‑dimensional manifold equipped with a Cartan vector field τ that encodes the total derivative along solutions of a differential system. The authors restrict to the case of a single Cartan field (ordinary diffieties) and give precise definitions of the order of functions and of differential operators.

The central contribution is the definition of an integrable infinitesimal symmetry δ: a vector field that not only commutes with τ (i.e.,