Continuous and discrete symmetries have played an important role in the theory of diffrerential equations and drove M.S.Lie in his development of Lie Group theory, or Transformation theory. Lie's motivation came, in part, from the work of E.Galois on algebraic equations which was also developed for differential equations by M.E.Vessiot and E.Picard. For modern treatments of Lie's approach, with applications, see books by L.V.Ovsiannikov and P.J.Olver.
Lie symmetry theory, as opposed to the abstract theory of Lie Groups, has become a very practical method in the toolbox of the applied mathematician. Part of the motivation behind the references cited below, however, is to relate these results to the abtract differential Galois theory of Picard and Vessiot. For modern treatments of the differential Galois theory see books by I.Kaplansky and A.Magid.
- Athorne, C., On the Lie symmetry algebra of a general ordinary differential equation, J. Phys. A 31 (1998) 6605-6614. MathSciNet Review
- Athorne, C., Symmetries of linear, ordinary differential equations, J. Phys. A 30 (1997) 4639-4649.MathSciNet Review
- Hartl, T. and Athorne, C., Solvable structures and hidden symmetries, J. Phys. A 27 (1994) 3463-3474.MathSciNet Review.