Lie group analysis of some new types of integrable differential equations (Q1312719)

The use of Lie groups is an established way of attempting to solve differential equations. See for instance [\textit{G. Bluman} and \textit{S. Kumei}, ``Symmetries and differential equations'', Springer, New York (1989; Zbl 0698.35001); \textit{P. J. Olver}, ``Applications of Lie groups to differential equations'', Springer, New York (1986; Zbl 0588.22001); \textit{H. Stephani}, ``Differential equations: Their solution using symmetries'', CUP, Cambridge (1989; Zbl 0704.34001)]. Essentially one seeks a point symmetry of the equations being studied and reduces the order or dimensionality by restricting to the invariants of the (local) symmetry group. The present author adds to a literature on second order linear and nonlinear ordinary differential equations studied via this method. He establishes a correspondence between the coefficients in a generic linear equation and those in the infinitesimal symmetry (vector field). By judicious choice of the latter he is then able to describe some integrable classes. This is a rather ad hoc, though traditional, way of proceeding and the results are necessarily of very restricted application.