In this lecture we discuss a class of homogeneous Finsler metrics of vanishing S-curvature on a (4n+3)-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature 1 among this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature 1 and vanishing S-curvature. They contain the canonical metric on S^{4n+3} of constant sectional curvature 1 and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.
