On Lie systems and Kummer-Schwarz equations

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer-Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne-Pinney and to the here defined generalised Kummer-Schwarz equations, which include several types of Kummer-Schwarz equations as particular cases. We demonstrate that all the above-mentioned equations related to the same Lie system on SL(2,R) can be integrated simultaneously, which retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.