The solutions of PIII(0, 0, 4, −4) on ℝ> 0 which take values in ℝ or in S 1 are related to the TERP structures which the second author had defined in [He03], motivated by [CV91, CV93, Du93], and which were studied subsequently in [HS07, HS10, HS11] [Mo11b, Sa05a, Sa05b] and other papers. They generalize variations of (polarized) Hodge structures. The concept of TERP(0) bundle is defined below in Definition 16.1. It is a TEP bundle with an additional real structure. It can be pure or not, and if it is pure, it can be polarized or not. A pure polarized TERP(0) bundle generalizes a polarized Hodge structure.