Exact solutions admitting isometry groups G_r ⫆ abelian G₃

Metrics admitting a minimal three dimensional Abelian isometry group, G₃ 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 (Formula Presented) G₃ with r=4, 6, 7 and 10, whereas the classes of metrics of type D admit G_r (Formula Presented) G₃ 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 G₃ or G₄. This classification also proves that the only non-null Einstein-Maxwell field admitting a maximal G₄ (Formula Presented) G₃ is the type D metric (6.7) which is of Segre type [(1, 1)(11)] and is isometric to the McVittie solution.