Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

Abstract

Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, i.e., classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of Rn+m, and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of R4.