$\mathcal{C}^{\infty}$-symmetries of distributions and integrability

TL;DR

The paper introduces $ abla$-symmetries for involutive distributions, enabling step-by-step integration of differential equations by solving successive integrable Pfaffian equations, extending classical symmetry methods.

Contribution

It generalizes the concept of symmetry for distributions, allowing new integrability techniques for differential equations based on $ abla$-structures.

Findings

New $ abla$-structures facilitate integrability of involutive distributions.

Provides a step-by-step method to solve higher-order ODEs.

Successfully applied to equations unsolvable by standard methods.

Abstract

An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to $\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years. These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental role in the integrability of the distribution: the knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution permits to find its integral manifolds by solving $k$ successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an $m$th-order ordinary differential equation by splitting the problem into $m$ completely integrable Pfaffian equations. This step-by-step integration procedure is applied to integrate completely several equations that cannot be solved by standard procedures.