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

DOI10.1016/J.JDE.2022.11.051zbMATH Open1518.34047arXiv2206.11737MaRDI QIDQ6402968

A. J. Pan-Collantes, Adrián Tonatiuh Ruiz, Juan Luis Romero, Concepción Muriel

Publication date: 23 June 2022

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.

Mathematics Subject Classification ID

Symmetries, invariants of ordinary differential equations (34C14) Explicit solutions, first integrals of ordinary differential equations (34A05) Applications of Lie algebras and superalgebras to integrable systems (17B80)

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