  The Einstein field equations for a class of irrotational non-orthogonally
transitive $G_{2}$ cosmologies are written down as a system of partial
differential equations. The equilibrium points are self-similar and can be
written as a one-parameter, five-dimensional, ordinary differential equation.
The corresponding cosmological models both evolve and have one-dimension of
inhomogeneity. The major mathematical features of this ordinary differential
equation are derived, and a cosmological interpretation is given. The
relationship to the exceptional Bianchi models is explained and exploited to
provide a conjecture about future generalizations.
