  We establish an extended version of the modified S\'{a}ez-Ballester (SB)
scalar-tensor theory in arbitrary dimensions whose energy momentum tensor as
well as potential are pure geometrical quantities. This scenario emerges by
means of two scalar fields (one is present in the SB theory and the other is
associated with the extra dimension) which widens the scope of the
induced-matter theory. Moreover, it bears a close resemblance to the standard
S\'{a}ez-Ballester scalar-tensor theory, as well as other alternative theories
to general relativity, whose construction includes either a minimally or a
non-minimally coupled scalar field. However, contrary to those theories, in our
framework the energy momentum tensor and the scalar potential are not added by
hand, but instead are dictated from the geometry. Concerning cosmological
applications, our herein contribution brings a new perspective. We firstly show
that the dark energy sector can be naturally retrieved within a strictly
geometric perspective, and we subsequently analyze it. Moreover, our framework
may provide a hint to understand the physics of lower gravity theories.
