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. This paper is written in a rather
pedagogical way so that it is suitable for teachers, undergraduate and
postgraduate students from physics and mathematics disciplines.
