Coupled Painlevé II systems in dimension four and the systems of type $A_4^{(1)}$

Abstract

We find and study a two-parameter family of coupled Painlevé II systems in dimension four, which can be obtained by a degeneration from the systems of type $A_4^{(1)}$. These systems are compared with other types of coupled Painlevé II systems from the viewpoint of the local index. We also give the phase spaces for these systems.