We consider a generalization of the well-known static axisymmetric vacuum solution of the Einstein equations in the case where the central naked singularity is also a source of some non-gravitational field or fields with the diagonal energy-momentum tensor (EMT). We find the condition for EMT components, which makes such a generalization possible. In the special case of massless field, when all components of the EMT have the same coordinate dependence, we obtain two generalized metrics and study the properties of these space-times, in particular, the type of singularities.
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