Initial value space of the fourth order Painlevé system with $A_5^{(1)}\times A_1^{(1)}$ symmetry

The initial value spaces of the Painlevé equations are proposed by Okamoto. They are symplectic manifolds in which the Painlevé equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the fourth order Painlevé system with affine Weyl group symmetry of type $A_5^{(1)}\times A_1^{(1)}$.