  Group theoretical methods are used to study the equations describing
\chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable
by inverse scattering techniques. On the other hand, these equations do share
some of the nice properties of soliton equations. Large families of explicit
analytical solutions are obtained in terms of elliptic functions. In special
cases, these periodic solutions reduce to localized ones, i.e., solitary waves.
All previously known explicit solutions are recovered, and many additional ones
are obtained
