Nonintegrability of coupled Painlevé systems with affine Weyl group symmetry of type \(A_4^{(2)}\) (Q6633087)

The Sasano system is the following system of equations\N\[\N\dot{x}=4xy-2\alpha_1+2zw \N\]\N\[\N\dot{y}=-2y^2-4x-2t-w \N\]\N\[\dot{z}=z^2-w+2y \N\]\N\[\N\dot{w}=-2zw-\alpha_0-2yw \N\]\NThis system can be considered as coupled Painleve systems.\N\NThe Sasano system admits Backlund transformations and the Weyl group \(W(A_4^{(2)})\) is realized by Backlund transformations.\N\NThe author considers parameters for which the Sasano system has a particular rational solution. It is proved that in this case it is not integrable by rational first integrals.