The paper is devoted to the problem of classifying the locally projectively Ricci-flat Rander metrics on the standard unit sphere.
Let \(M\) be an \(n\)-dimensional smooth manifold, \(dV\) - the volume form on \(M, \boldsymbol{G}\) -a spray (i.e. a projective Finsler geometry) on \(M, \boldsymbol{S}\) - the S-structure of \((\boldsymbol{G}, dV), \boldsymbol{Y}\) - the canonical vertical vector field on the tangent bundle \(TM\), \(\hat{{\boldsymbol{G}}} := G + \frac{2{\boldsymbol{S}}}{n+1} {\boldsymbol{Y}}, \boldsymbol{PRic}_{{\boldsymbol{G}},dV}=\boldsymbol{PRic}_{\hat{{\boldsymbol{G}}}}, F=\alpha+\beta\)- a locally projective flat Rander metric, i.e. the sum of a Riemannian metric \(\alpha\) and a 1-form \(\beta\) regarded as a smooth function on \(TM\), \((M,F,dV)\) - a Finsler measure space. As the main result of the paper, in the case of a standard sphere \(({\boldsymbol{S}}^n,\alpha)\) and the Busemann-Hausdorff \(S\)-curvature \({\boldsymbol{S}}_{BH}= (n+1)\frac{< {\boldsymbol{v}},x>}{2\sqrt{1 - < {\boldsymbol{v}},i(x)>^2}}F\) with respect to the Busemann-Hausdorff volume form \({\boldsymbol{d}}V_{BH}\), the Rander metric \(F(x,y) = \alpha(x,y) - \frac{< {\boldsymbol{v}},i_*y>}{\sqrt{1-< {\boldsymbol{v}}, i(x)>^2}}\), for \(\vert \vert {\boldsymbol{v}}\vert \vert <1\), is projectively Ricci-flat on the outside of the intersetion of \({\boldsymbol{S}}^n\) with the hyperplane \(\Pi_{\boldsymbol{v}}\subset \mathbb{R}^{n+1}\) through the origin in \(\mathbb{R}^{n+1}\) with normal vector \(\boldsymbol{v}\) and in that case \(\boldsymbol{{PRic}}_{({\boldsymbol{G}}_F, e^{-\phi}dV_\alpha)} = 0\), where \(\phi(x) = \ln\vert < {\boldsymbol{G}}, i(x)>\vert ^{n+1}\) (Theorems 1.1, 1.2, 5.1, Proposition 5.2).