1 M ay 2 00 5 COUPLED PAINLEVÉ II SYSTEMS IN DIMENSION 4 AND THE SYSTEMS OF TYPE

H = xy(2y − x− 2t)− 2β1y − β2x+ zw(2w − z − 2t)− 2β3w − β4z + 4yzw = HIV (x, y, t; β1, β2) +HIV (z, w, t; β3, β4) + 4yzw. Here, x, y, z and w denote unknown complex variables, and β1, β2, β3 and β4 are complex parameters. It is well-known that PIV has a confluence to the second Painlevé equation PII , where two accessible singularities come together into a single singularity. This suggests the possibility that there exists a procedure for searching for higher order versions of Painlevé II, by using Takano’s description of the confluence process from PIV to PII for the coordinate systems (x, y) and (z, w), respectively. In this vein, the goal of this work is to find a fourth-order version of the Painlevé II equation. We present a 2-parameter family of fourth-order algebraic ordinary differential equations which can be considered as coupled Painlevé II systems in dimension 4, and which is given as follows: