  Using a modified version of Weinstein's argument for constrained minimization
in nonlinear dispersive equations, we prove existence of solitary waves in
fully nonlocally nonlinear equations, as long as the linear multiplier is of
positive and slightly higher order than the Coifman-Meyer nonlinear multiplier.
It is therefore the relative order of the linear term over the nonlinear one
that determines the method and existence for these types of equations. In
analogy to Korteweg-De Vries-type equations and water waves in the capillary
regime, smooth solutions of all amplitudes can be found. We consider two
structural types of symmetric Coifman-Meyer symbols $n(\xi-\eta,\eta)$, and
show that cyclical symmetry is necessary for the existence of a functional
formulation. Estimates for the solution and wave speed are given as the
solutions tend to the bifurcation point of solitary waves.
