Ordinary differential equations on rational elliptic surfaces

Abstract

Corresponding to Oguiso-Shioda's classification of rational elliptic surfaces, we give 2nd order algebraic ordinary differential equations which can be solved by elliptic functions, in the form of the Hamiltonian system. There is a criterion for determining the types of rational elliptic surfaces from given biquadratic Hamiltonian systems. We also discuss about Bäcklund transformations which is different type from transformations that appear in a study of the Painlevé equations. © Springer-Verlag London 2013.