On the generalizations of the Kummer-Schwarz equation (Q1985811)

The paper under review proposes the study of the symmetries of the following four differential equations: \(S(y)=f(y)(y^{\prime })^2\), \(S(y)=f(y)\), \(S_n(y)=f(y)(y^{\prime })^2\), \(S_n(y)=f(y)\), where \(S\) is the usual Schwarzian derivative \(S(y)=\frac{y^{\prime \prime \prime }}{y^{\prime }}-\frac{3}{2}\left(\frac{y^{\prime \prime }}{y^{\prime }}\right)^2\) and \(S_n\) is a deformed Schwarzian derivative \(S_n(y)=\frac{y^{\prime \prime \prime }}{y^{\prime }}+n\left(\frac{y^{\prime \prime }}{y^{\prime }}\right)^2\) with \(n\in \mathbb{R}\) not necessarily integer. It is proved that the symmetry algebra of the first equation above, called \textit{non-homogeneous Kummer-Schwarz equation}, is \(sl(2, \mathbb{R})\oplus sl(2, \mathbb{R})\) which is six-dimensional. Another main result is that this equation is the most general third-order ODE admitting the vector fields \(X_1=\frac{\partial }{\partial x}\), \(X_2=x\frac{\partial }{\partial x}\), \(X_3=x^2\frac{\partial }{\partial x}\), from the representation of the Lie algebra \(sl(2, \mathbb{R})\), as Lie point symmetries.