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