A Cubic Hamiltonian System with Meromorphic Solutions

Abstract

A Hamiltonian system in two dependent variables is presented with properties analogous to the Hamiltonian systems associated with the six Painleve equations. Its solutions are meromorphic functions in the complex plane having only simple poles with three possible residues given by the third roots of unity. Like the Painleve equations $$P_\mathrm{II}$$ – $$P_\mathrm{VI}$$ the system has families of rational solutions that can be obtained by applying Backlund transformations to certain seed solutions.