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In this paper, we bring together the five-dimensional Saez-Ballester~(SB)scalar-tensor theory [1] and the induced-matter-theory~(IMT) setting [2], toobtain a modified SB theory (MSBT) in four dimensions. Specifically, by usingan intrinsic dimensional reduction procedure into the SB field equations infive-dimensions, a MSBT is obtained onto a hypersurface orthogonal to the extradimension. This four-dimensional MSBT is shown to bear distinctive new featuresin contrast to the usual corresponding SB theory as well as to IMT and theModified Brans-Dicke Theory (MBDT)~\cite{RFM14}. It should be emphasized thatthe herein appealing solutions can emerge solely from the geometricalreductional process, from presence also of extra dimension(s) and not from anyad-hoc matter either in the bulk or on the hypersurface.  Subsequently, we apply the herein MSBT to cosmology and consider an extendedspatially flat FLRW geometry in a five-dimensional vacuum space-time. Afterobtaining the exact solutions in the bulk, we proceed to construct, by means ofthe MSBT setting, the corresponding dynamic, on the four-dimensionalhypersurface. More precisely, we obtain the (SB) components of the inducedmatter, including the induced scalar potential terms. We retrieve two differentclasses of solutions. Concerning the first class, we show that the MSBT yieldsa barotropic equation of state for the induced perfect fluid. We theninvestigate vacuum, dust, radiation, stiff fluid and false vacuum cosmologiesfor this scenario and contrast the results with those obtained in the standardSB theory, IMT and BD theory. Regarding the second class solutions, we showthat the scale factor behaves similar to a de Sitter (DeS) model. However, inour MSBT setting, this behavior is assisted by non-vanishing induced matterinstead, without any a priori cosmological constant.

Modified Saez-Ballester scalar-tensor theory from 5D space-time