Evolution of the concept of λ–symmetry and main applications

Resumo

For ordinary differential equations, the genesis of the concept of C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263743/15e65f54-7536-469c-b2b1-9a6228e9465d/content/eq1700.tif"/> ∞–symmetry (or λ–symmetry), several of its equivalent formulations and some of its relationships with other concepts of symmetry are considered. A description of the main applications of the λ–symmetries theory, including its role in linearisation problems and the search of integrating factors, first integrals and Jacobi last multipliers, is provided. Some connections with other integration or reduction procedures are also explored.A review of the generalisations of the concept of λ–prolongation of a vector field and the extensions of the concept of λ–symmetry outside the framework of ordinary differential equations has also been performed. Several extensions and open problems of interest in this theory are discussed.