Differ Equ Dyn Syst DOI 10.1007/s12591-016-0327-5 ORIGINAL RESEARCH Algebraic and Singularity Properties of a Class of Generalisations of the Kummer–Schwarz Equation R. Sinuvasan 1 · K. M. Tamizhmani 1 · P. G. L. Leach 2,3 © Foundation for Scientific Research and Technological Innovation 2016 Abstract The Kummer–Schwarz Equation, 2 y ′ y ′′′ − 3 y ′′ 2 = 0, (the prime denotes differ- entiation 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 sym- metry 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. Keywords Kummer–Schwarz · Symmetries · Singularities · Integrability Mathematics Subject Classification 34A05 · 34A34 · 34C14 · 22E60 Introduction The Kummer–Schwarz Equation [11], 2 y ′ y ′′′ − 3 y ′′ 2 = 0, (1) is notable amongst the class of third-order ordinary differential equations due to its properties as a differential equation apart from its well-known connection to the Schwarzian Derivative. B R. Sinuvasan rsinuvasan@gmail.com P. G. L. Leach leach@ucy.ac.cy 1 Department of Mathematics, Pondicherry University, Kalapet 605 014, India 2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, Republic of South Africa 3 Department of Mathematics, Institute for Systems Science, Durban University of Technology, POB 1334, Durban 4000, Republic of South Africa 123