Algebraic and Singularity Properties of a Class of Generalisations of the Kummer–Schwarz Equation

The Kummer–Schwarz Equation, $$2 y'y''' - 3 y''{}^2 = 0$$, (the prime denotes differentiation with respect to the independent variable x) is well known from its connection to the Schwartzian Derivative and in its own right for its interesting properties in terms of symmetry and singularity. We examine a class of equations which are a natural generalisation of the Kummer–Schwarz Equation and find that the algebraic and singularity properties of this class of equations display an attractive set of patterns. We demonstrate that all members of this class are readily integrable.