Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$

Abstract

We find a four-parameter family of coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$. This is the first example which gave higher order Painlevé type systems of type $E_6^{(2)}$. We study its symmetry and holomorphy conditions.

6 pages, 1 figure