Symmetry algebras of third-order ordinary differential equations

The main result of this paper is the complete classification of the third-order ordinary differential equations according to their symmetries. The same classification was done for second order by Tresse ["Dé termination des invariants ponctuels de l'équation différentielle ordinaire du second ordre y"=ω(x,y,y′)," Gekrönte Preisschrift, Hirzel, Leipzig (1896)], and recently for arbitrary order linear ordinary differential equations. The sections preceding the classification consist of a brief description of the concepts and methods along the lines of Krause and Michel [Lecture Notes Phys. 382, 251 (1991)]. These sections also contain some definitions and a table listing the prolongations of a few vector fields. Finally, two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders.