V. F. Zaitsev
St.Petersburg State University, Russia
At the end of the 19th century S. Lie offered a group approach for integration of ordinary differential equations (ODE) generalizing known algorithms of the first generation. These algorithms are a collection of empirical methods forming the basis of E. Kamke's reference book and were in use until the middle of the 20th century. A collection of algorithms of the second generation completely formed to date provides means to find symmetries of ODE (Lie groups, discrete groups, and first integrals) and leads to integration of an equation in the analytic form when equation possesses a sufficient number of symmetries.