Symmetries of distributions and quadrature of ordinary differential equations (Q1181252)

The authors present a very simple geometrical exposition of the most elementary principles of \(S\). Lie's theory of explicit integration by quadratures of ordinary differential equations. Besides interesting historical comments and a very clear proof of the famous Lie's general criterion (the existence of a solvable symmetry group), the article involves several non-trivial examples (e.g., equations \(y''=y'+f(y)\) which possess two-dimensional symmetry, explicit solutions of partial differential equations, the equation \(ay^ 2y'''+byy'y''+cy'{}^ 3=0)\). All fundamental concepts (including differential equations in terms of jets, outer and inner symmetries, finite-type systems and prolongations) are recalled. Unfortunately, the more advanced Cartan's and Vessiot's papers, the role of pseudogroups, and the recent Bäcklund symmetries are passed over in silence.