  We construct in an explict way the soliton equation corresponding to the
affine Kac--Moody Lie algebra $G_2^{(1)}$ together with their bihamiltonian
structure. Moreover the Riccati equation satisfied by the generating function
of the commuting Hamiltonians densities is also deduced. Finally we describe a
way to deduce the bihamiltonian equations directly in terms of this latter
functions
