  We generalize the method used by M{\ae}hlen & Seth [17] used to prove the
existence of small-amplitude asymmetric solutions to the capillary-gravity
Whiham equation, so that it can be applied directly to a class of similar
equations. The purpose is to prove or disprove the existence of asymmetric
waves for the water wave problem or other model equations for water waves. Our
main result in this paper is a theorem that gives both necessary and sufficient
conditions for the existence of small-amplitude periodic asymmetry solutions
for this class of equations. The result is then applied to an infinite depth
capillary-gravity Whitham equation and an infinite depth capillary-gravity
Babenko equation to show the nonexistence of small-amplitude waves for these
equations. This example also highlights the similarities between these
equations suggesting the potential existence of small-amplitude asymmetric
waves for the finite depth capillary-gravity Babenko equation.
