Cohomogeneity-one Einstein metrics on closed $4$-manifolds

We classify the Einstein metrics on closed 4-manifolds whose isometry group has dimension at least 3. Consequently, we show that the cohomogeneity-one Einstein metrics on a closed $4$-manifold are locally symmetric or homothetic to the Page metric on ${\mathbf{C}\mathbf{P}}^2\mathrm{♯}{\overline{\mathbf{C}\mathbf{P}}}^2$.