  We consider the two-dimensional capillary water waves with nonzero constant
vorticity in infinite depth. We first derive the Babenko equation that
describes the profile of the solitary wave. When the velocity $c$ is close to a
critical velocity and a sign condition involving the physical parameters is
met, the Babenko equation can be reduced to the stationary focusing cubic
nonlinear Schr\"odinger equation plus perturbative error. We show the existence
of a critical value of a dimensionless physical parameter below which at least
two families of velocities satisfy the focusing condition and above which only
one does. This gives the existence of small-amplitude solitary wave solutions
for the water wave system with constant vorticity.
