Constant scalar curvature metrics on Hirzebruch surfaces

We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our construction is reduced to an ordinary differential equation called Duffing equation. An ODE for Bach-flat metrics on Hirzebruch surfaces with large isometry group is also derived.