This paper is concerned with the rheometrical characterization of a family of liquids based on the S1 fluid which is currently the object of an international research programme. The programme has, as primary objective, the study of how a solvent influences the rheological properties of a polymer solution for as broad a range as possible of the solvent concentration. S1 is a specific solution of polyisobutylene (PIB) in a solvent which consists of a mixture of polybutene oil and Decalin. In this paper, a family of solutions having different concentrations of PIB is studied. Conventional rheometrical data is presented for the solutions and these are fitted to useful but relatively simple equations. Although the usual limiting relationships that are expected for continuous media are shown to be satisfied, the empirical Cox-Merz rule is found to be violated. The characterization was complicated by the occurrence of an apparent shear-thickening phenomenon which was studied in detail and explained. Also, the now familiar instability that often occurs in rotary shear flows was shown to occur, as expected, at the same Deborah number for each of the solutions studied.