General forms of the metric for spacetimes with axial symmetry

We find all the general forms which a spacetime metric can take, when this spacetime obeys a discrete or continuous axial symmetry, whether it is rota ting or not. If the spacetime is not symmetrical under a reflection through a plane orthogonal to its symmetry axis, there are two different classes: the first class corresponds to a discrete symmetry of order two, all the ot her symmetries, be they discrete or continuous, are linked to the same gene ral metric form. If the spacetime does possess this reflection symmetry, we have four classes of metrics. The three classes having discrete admit a no nstatic metric.