  We consider the nonlinear problem of steady gravity-driven waves on the free
surface of a two-dimensional flow of an incompressible fluid (say, water). The
flow is assumed to be unidirectional of finite depth and the water motion is
supposed to be rotational. Our aim is to verify the Benjamin--Lighthill
conjecture for flows whose total head (Bernoulli's constant) is close to the
critical one; the latter is determined by the vorticity distribution so that no
horizontal shear flows exist for smaller values of the total head.
