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Using center manifolds and topological degree theory, we construct a newfamily of complete, $SU(2)$-invariant and steady gradient Ricci solitons on thefour-dimensional non-compact cohomogeneity one manifold with group diagram$\mathbb{Z}_4\subset U(1)\subset SU(2)$. We also provide simpler constructionsof the existing $U(2)$-invariant steady and complete gradient solitons on thecohomogeneity one manifolds with group diagrams $\mathbb{Z}_n\subsetU(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton'snon-collapsed solitons for $n\ge 3$.

$SU(2)$-invariant steady gradient Ricci solitons on four-manifolds