Incompressible Euler fluids on compact cohomogeneity one manifolds

Let $(M,\mathsf{g})$ be a connected and compact Riemannian manifold admitting an isometric action by a compact Lie group $G$ whose principal orbits have codimension one. We show that any $G$-invariant, smooth, and divergence-free vector field $u_0$ on $(M,\mathsf{g})$ initiates a $G$-invariant time-varying velocity-pressure pair $(u,p)$ which has time interval $\mathbb{R}$, is smooth, and solves the incompressible Euler fluid equations.