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It is generated by two non-commuting subgroups: the Matzner–Misner group (after Richard Alfred Matzner and Charles W. Misner) of linear combinations (with constant coefficients) of the two Killing vector fields associated with the spacetime's axisymmetry and stationarity, and the Ehlers group.
A more general subclass consists of the axisymmetric pp-waves, which in general have a two dimensional Abelian Lie algebra of Killing vector fields.
Schwarzschild has 4 independent Killing vector fields, including a timelike one, and thus is a static metric, while Vaidya has only 3 independent Killing vector fields regarding the spherical symmetry, and consequently is nonstatic.