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We consider the two dimensional pure gravity water waves with nonzeroconstant vorticity in infinite depth, working in the holomorphic coordinatesintroduced by Hunter, Ifrim, and Tataru. We show that close to the criticalvelocity corresponding to zero frequency, a solitary wave exists. We use afixed point argument to construct the solitary wave whose profile resembles arescaled Benjamin-Ono soliton. The solitary wave is smooth and has anasymptotic expansion in terms of powers of the Benjamin-Ono soliton.

Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case