Exact solutions admitting isometry groups G r supE Abelian G 3

Metrics admitting a minimal three dimensional Abelian isometry group, G 3 are classified according to their Petrov types and metrics, giving all type O and D metrics explicitly, without imposing a source condition. The corresponding maximal Lie algebras for these metrics are obtained and identified as well. The type O metrics admit a maximal ${G}_{r}\supset $ G 3 with r = 4, 6, 7 and 10, whereas the classes of metrics of type D admit ${G}_{r}\supseteq $ G 3 with r = 3, 4, 5 and 6 as the maximal isometry groups. Type O metrics with a perfect fluid source are then found explicitly and are shown to admit a maximal G r with r = 4, 7 and 10. Type D perfect fluid metrics are found explicitly which admit either a maximal G 3 or G 4. This classification also proves that the only non-null Einstein-Maxwell field admitting a maximal ${G}_{4}\supset $ G 3 is the type D metric (6.7) which is of Segre type [(1, 1) (1 1)] and is isometric to the McVittie solution.