In this monograph we are interested in the Painlevé III(D 6) equation of type (α, β, γ, δ) = (0, 0, 4, −4) and in the isomonodromic families of P3D6 bundles which are associated to it in [FN80, IN86, FIKN06, Ni09]. These P3D6 bundles are special and can be equipped with rich additional structure. We shall develop this structure in two steps in Chaps. 6 and 7 The most important part is the TEP structure below. In Chap. 7 it will be further enriched to a TEJPA structure. Isomonodromic families of P3D6-TEJPA bundles will correspond to solutions of the equation PIII(0, 0, 4, −4) (Theorem 10.3).