Coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of types $A_4^{(2)}$ and $A_1^{(1)}$

Abstract

We find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_4^{(2)}$. For a degenerate system of $A_4^{(2)}$ system, we also find a one-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_1^{(1)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new. Moreover, we find a one-parameter family of partial differential systems in three variables with $W(A_1^{(1)})$-symmetry. We show the relation between its polynomial Hamiltonian system and an autonomous version of the system of type $A_1^{(1)}$.