Lambda and mu-symmetries

Symmetry analysis is a standard and powerful method in the analysis of differential equations, and in the determination of explicit solutions of nonlinear ones. It was remarked by Muriel and Romero [10] (see also the work by Pucci and Saccomandi [14]) that for ODEs the notion of symmetry can be somehow relaxed to that of lambda-symmetry (see below), still retaining the relevant properties for symmetry reduction and hence for the construction of explicit solutions. Their work was presented at SPT2001 [11], raising substantial interest among participants. Here I report on some recent work [4, 6, 7] which sheds some light on “lambda-symmetries”, and extends them to PDEs – and systems thereof – as well; as the central objects here are not so much the functions λ, but some associated one-forms μ, these are called “mu-symmetries”. The work reported here was conducted together with Giampaolo Cicogna and Paola Morando; I would like to thank them, as well as other friends (J.F. Cariñena, G. Marmo, M.A. Rodŕıguez) with whom I discussed these topics in the near past. It is also a pleasure to thank C. Muriel and G. Saccomandi for privately communicating their work on λ-symmetries and raising my interest in the topic.