  The Lax pair representation in Fourier space is used to obtain a list of
integrable scalar evolutionary equations with quadratic nonlinearity. The
famous systems of this type such as KdV, intermediate long-wave equation (ILW),
Camassa-Holm and A. Degasperis systems are represented in this list. Some new
systems are obtained as well. The generalizations on two-dimensional and
discrete systems are discussed.
