On the equivalence between S\'aez-Ballester theory and Einstein-scalar field system

Here we discuss a topic that comes up more often than expected: A same theory or theoretical model arises in two different presentations which are assumed to be actually different theories so that these are independently developed. Sometimes this leads to an unwanted doubling of the results. In this paper we illustrate this issue with the example of two apparently different gravitational theories: (i) the (minimally coupled) Einstein-massless-scalar system and (ii) the S\'aez-Ballester theory. We demonstrate that the latter is not a scalar-tensor theory of gravity, as widely acknowledged. Moreover, S\'aez-Ballester theory is identified with the Einstein-massless-scalar theory. As illustrations of this identification we show that several known solutions of S\'aez-Ballester theory are also solutions of the Einstein-massless-scalar system and viceversa. Cosmological arguments are also considered. In particular, a dynamical systems-based demonstration of the dynamical equivalence between these theories is given. The study of the asymptotic dynamics of the S\'aez-Ballester based cosmological model shows that there are not equilibrium points which could be associated with accelerated expansion, unless one includes a cosmological constant term or a self-interacting scalar field. This is a well-known result for cosmological models which are based in the Einstein-self-interacting-scalar theory, also known as quintessence.