Petrov classification of vacuum axisymmetric space-times

Using the null bivector approach, Petrov classification is studied for axisymmetric vacuum space-times with orthogonally transitive Killing vectors. It is shown that the equation on which the classification is based is biquadratic. This excludes that any such space-time can be type III. The only type-N manifolds are the radiative solutions considered by Hoffman. Van Stockum solutions are the most general type-II solutions. Degenerate Weyl solutions and Kinnersley solutions cover typeD. Stationary solutions with functional dependence of the potential are then examined. It is found that, except for special cases, Papapetrou and Lewis solutions are algebraically general.