Axially symmetric and static solutions of Einstein equations with self-gravitating scalar field

Abstract: The exact axisymmetric and static solution of the Einstein equations coupled to axisymmetric and static gravitating scalar (or phantom) field is presented. The spacetimes modified by the scalar field are explicitly given for the so called $\gamma$-metric and Erez-Rosen metric with quadrupole moment $q$, influence of the additional deformation parameters $\gamma_$ and $q_$ generated by the scalar field is studied. It is shown that the null energy condition is satisfied for the phantom field, but it is not satisfied for the standard scalar field. The test particle motion in the both modified $\gamma$-metric and Erez-Rosen quadrupole metric is studied; the circular geodesics are determined, and near-circular trajectories are explicitly presented for characteristic values of the spacetime parameters. It is also demonstrated that the parameters $\gamma_$ and $q_$ have no influence on the test particle motion in the equatorial plane.