Recently, it has been shown that the class of bodies that can be called elastic is far larger than classical Cauchy and Green elastic bodies. In order to assess the usefulness and viability of this new class of elastic bodies, it is necessary to evaluate solutions to typical initial-boundary-value problems and to see if the predictions of these models agree with observed phenomena. With this in mind, we studied members of the new class of elastic bodies within the context of a physically meaningful boundary-value problem, namely that of an infinite slab subject to a state of simple shear. Even though the problem that we consider seems very simple, we found a richness of solutions that was quite surprising. We used a non-standard semi-inverse method and looked for a special form for the solution of the problem. We found that, depending on the choice of models for the new class, this simple problem presents the most interesting outcomes: the possibility that there exists only one solution to the problem, the possibility that there exists more than one solution to the problem, and the possibility that there exists no solution, of the form sought within the context of the semi-inverse form to the problem. The non-uniqueness that is encountered is different from the traditional sense, as will be explained in the text. We were able to establish several exact solutions, which in addition to being useful solutions can also be used to check the efficacy of numerical solutions to more complicated problems. We also find the interesting possibility that solutions exhibit pronounced layers where the strain gradient is significantly larger than in the rest of the domain.