  The only known example of collapsed three-dimensional complete gradient
steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$,
the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$
with the product metric. R. Hamilton has conjectured that there should exist a
family of collapsed positively curved three-dimensional complete gradient
steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton.
In this paper, we make the first initial progress and prove that the
infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix
A, we show that the 3D cigar soliton is the unique complete nonflat gradient
steady Ricci soliton in dimension three that admits two commuting Killing
vector fields.
