2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65141We present transformations relating the third transcendent of Painlevé with parameter sets located at the corners of the Weyl chamber for the symmetry group of the system, the affine Weyl group of the root system $ B^{(1)}_2 $, to those at the origin. This transformation entails a scaling of the independent variable, and implies additive identities for the canonical Hamiltonians and product identities for the $τ$-functions with these parameter sets.AMSLatex with Proc. Amer. Math. Soc. macrosClassical Analysis and ODEsExactly Solvable and Integrable Systems34M55;33E17;20F55text