Coupled Painlevé II Systems in Dimension 4 and the systems of type ${A_4}^{(1)}$

Abstract

We find and study coupled Painlevé II systems in dimension 4, which can be obtained by a degeneration from the systems of type ${A_4}^{(1)}$. We compare these systems with other types of coupled Painlevé II systems from the viewpoint of the local index. We also give the phase spaces for these systems.