In the previous four chapters we discussed metrics which admit two commuting space-like Killing vector fields. In this chapter we deal with stationary axisymmetric spacetimes where one of the two Killing fields is time-like. These spacetimes have been investigated for a long time due to the possibility of describing the gravitational fields of compact astrophysical sources. The field equations for the relevant metric tensor components are now elliptic rather than hyperbolic as in the nonstationary case but the solutions can be formally related via complex coordinate transformations. In section 8.1 we again formulate the ISM, but in this case, because of the different ranges of the coordinates, some of the previous expressions become much simpler. In section 8.2 the general n-soliton solution is explicitly constructed in this axisymmetric context. In section 8.3 the Kerr, Schwarzschild and Kerr–NUT solutions are constructed as simple two-soliton solutions on the Minkowski background. The asymptotic flatness of the general n-soliton solution is discussed in section 8.4 and we show that asymptotic flatness can always be imposed by certain restrictions on the soliton parameters; the resulting spacetimes can be interpreted as a superposition of Kerr black holes on the symmetry axis. In section 8.5 we discuss the diagonal metrics (static Weyl class). In this case the soliton metrics contain many well known static solutions and some generalized soliton solutions can be constructed as in the previous chapters; a few particularly interesting solutions are considered in some detail.