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We study the non Ricci flat gradient steady K\"{a}hler Ricci soliton withnon-negative Ricci curvature and weak integrability condition of the scalarcurvature $S$, 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 compactK\"{a}hler Ricci flat manifold respectively. As an application, we prove thatany non Ricci flat gradient steady K\"{a}hler Ricci soliton with $Ric\geq 0$,together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ musthave universal covering space isometric to $\Sigma\times\mathbb{C}^{n-1-k}\times N^k$.

Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature