  The extended scalar-tensor and vector-tensor theories admit black hole
solutions with the nontrivial profiles of the scalar and vector fields,
respectively. The disformal transformation maps a solution in a class of the
scalar-tensor or vector-tensor theories to that in another class, and hence it
can be a useful tool to construct a new nontrivial solution from the known one.
First, we investigate how the stationary and axisymmetric solutions in the
vector-tensor theories without and with the $U(1)$ gauge symmetry are
disformally transformed. We start from a stationary and axisymmetric solution
satisfying the circularity conditions, and show that in both the cases the
metric of the disformed solution in general does not satisfy the circularity
conditions. Using the fact that a solution in a class of the vector-tensor
theories with the vanishing field strength is mapped to that in a class of the
shift-symmetric scalar-tensor theories, we derive the disformed stationary and
axisymmetric solutions in a class of these theories, and show that the metric
of the disformed solutions does not satisfy the circularity conditions if the
scalar field depends on the time or azimuthal coordinate. We also confirm that
in the scalar-tensor theories without the shift symmetry, the disformed
stationary and axisymmetric solutions satisfy the circularity conditions.
Second, we investigate the disformal transformations of the stationary and
axisymmetric black hole solutions in the generalized Proca theory with the
nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor
theory with the nonminimal derivative coupling to the Einstein tensor, the
Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field
theory. We show that the disformal transformations modify the causal properties
of the spacetime.
