The method of reduction of order and linearization of the two-dimensional Ermakov system

We present the general form of the system of second-order ordinary differential equations invariant under a representation of the Lie algebra sl(2, R) and show that a considerable simplification is achieved using a well-known Kummer-Liouville transformation. We show that the system can be reduced to a combination of linear second-order ordinary differential equations and a conservation law. The reduction makes the determination of the complete symmetry group of the standard Ermakov system an easier task than earlier reported (J. Nonlinear Math. Phys. 2005; 12:305-320). The reduced system is equivalent to the reduction of the Kepler problem under a further constraint.