  The Chern-Simons amended gravity theory appears as a low-energy effective
theory of string theory. The effective theory includes an anomaly-cancelation
correction to the Einstein-Hilbert action. The Chern-Simons expression consists
of the product $\varphi R \tilde R $ of the Pontryagin density $R \tilde R $
with a scalar field $\varphi$, where the latter is considered as a background
field (dynamical construction or non-dynamical construction). Many different
solutions to Einstein's general relativity continue to be valid in the amended
theories. The Kerr metric is, however, considered an exceptional case that
raised a search for rotating black hole solutions. We generalize the solution
presented in Phys. Rev. D \textbf{77}, 064007 (2008) by allowing the potential
$V$ to have a non-vanishing value and we discuss three different cases of the
potential, that is, $V=\mathrm{const.}$, $V\propto \varphi$, and $V\propto
\varphi^2$ cases. The present study presents, for the first time, novel
solutions prescribing rotating black holes in the frame of the dynamical
formulation of the Chern-Simons gravity, where we include a potential and
generalize the previously derived solutions. We derive the solutions in the
slow-rotation limit, where we write the parameter of the slow-rotation
expansion by $\varepsilon$. These solutions are axisymmetric and stationary and
they give a distortion of the Kerr solution by a dipole scalar field. Moreover,
we investigate that the rectification to the metric behaves in the inverse of
the fourth order of radial distance from the center of the black hole when
$V\propto \varphi$. This suggests that any meaningful limits from the
weak-field experiments could be passed.
