One form deformation of sprays

In this paper, we introduce the notion of one form deformation of sprays. The metrizability of the new spray, when the background spray is flat, is characterized. Therefore, we obtain new projectively flat metrics of constant flag curvature $1$. Moreover, these new metrics are not, generally, isometric to the Klein metric via affine transformations. New solutions for Hilbert’s fourth problem are obtained and constructed. Various examples are discussed and studied.