  We study the non Ricci flat gradient steady K\"{a}hler Ricci soliton with
non-negative Ricci curvature and weak integrability condition of the scalar
curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$,
and show that it is a quotient of $\Sigma\times \mathbb{C}^{n-1-k}\times N^k$,
where $\Sigma$ and $N$ denote the Hamilton's cigar soliton and some compact
K\"{a}hler Ricci flat manifold respectively. As an application, we prove that
any non Ricci flat gradient steady K\"{a}hler Ricci soliton with $Ric\geq 0$,
together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ must
have universal covering space isometric to $\Sigma\times
\mathbb{C}^{n-1-k}\times N^k$.

Differential Geometry