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

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.

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Cited by in corpus (2)

• Integration of differential equations by ${\mathcal{C}}^{\infty }$-structures
• ${\mathcal{C}}^{\infty }$-structures in the integration of involutive distributions