Independent components of the curvature tensor for spacetimes with axial symmetry

We find a set of independent components for the Weyl and the Ricci parts of the curvature tensor of every spacetime with discrete or continuous axial symmetry, whether it is rotating or not. This set of independent components holds everywhere in the spacetime. If the spacetime is not symmetrical und er a reflection through a plane orthogonal to its symmetry axis, the curvat ure tensor belongs to one of two classes. The first class corresponds to a discrete symmetry of order two. The second class includes all the other sym metries, be they discrete or continuous. If the spacetime does possess this reflection symmetry, we have five classes. The first three classes corresp ond to discrete symmetries of order one, two and four. The fourth class inc ludes all the other discrete symmetries. The fifth class corresponds to the continuous symmetry. For each of these classes, we give the relations from which follow all components of the curvature tensor.