In this paper dl vacuum metrics with a four-parameter homothety group containing a threpdimensional Abelian subpoup are found. These include the Kssner family of metrics, metriw related to the Kaener family by complex transformations (which include a family of vacuum metrics found by Lewis in 1932 and a metric found by Petmv in 1962 to be the only vacuum metric which admits a simplytransitive fou~parameta isometry group), together with algebraic+ degenerate van Stodnun, ppwave and plane-wave metrics. Apart from the plan,wave metric, all the 0th5 metric6 are written here as one general family.
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