Derivation of Painlevé type system with $D_4^{(1)}$ affine Weyl group symmetry in a self-similarity limit

We show how the zero-curvature equations based on a loop algebra of $D_4$ with a principal gradation reduce via self-similarity limit to a polynomial Hamiltonian system of coupled Painlevé III models with four canonical variables and $D_4^{(1)}$ affine Weyl group symmetry.