Vainshtein screening for slowly rotating stars

We study the Vainshtein mechanism in the context of slowly rotating stars in scalar-tensor theories.
While the Vainshtein screening is well established for spherically symmetric spacetimes, we introduce a slow rotation of the source and examine its validity in the axi-symmetric case.
We solve for the frame-dragging function accounting for the slow rotation and show that the deviations from the general relativity (GR) solution are screened in the weak-field approximation, meaning the solution for the frame-dragging function is the same as GR to leading order.
This suggests that the Vainshtein mechanism in general can be extended to slowly rotating stars and it works in a similar way to the spherically symmetric case. We also discuss relativistic stars, and show that in some theories the frame-dragging function is the same as GR in vacuum, meaning that the screening is perfect outside the star.