Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface

uid surface of Stokes wave into the real line with uid domain mapped into the lower complex half-plane. There is one square root branch point per spatial period of Stokes located in the upper complex half-plane at the distance vc from the real axis. The increase of Stokes wave height results in approaching vc to zero with the limiting Stokes wave formation at vc = 0: The limiting Stokes wave has 2=3 power law singularity forming 2 =3 radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but nonzero vc making the limit of zero vc highly nontrivial. That limit is addressed by crossing a branch cut of square root into the second and higher sheets of Riemann surface to nd coupled square root singularities at the distances vc from the real axis at each sheet. The number of sheets is innite and the analytical continuation of Stokes wave into all these sheets is found together with the series expansion in half-integer powers at vc singular points within each sheet. It is conjectured that non-limiting Stokes wave at the leading order consists of the innite number of nested square root singularities. These nested square roots form 2=3 power law singularity of the limiting Stokes wave as vc vanishes.