  An interesting deformation of the Jackiw-Teitelboim (JT) gravity has been
proposed by Witten by adding a potential term $U(\phi)$ as a self-coupling of
the scalar dilaton field. During calculating the path integral over fields, a
constraint comes from integration over $\phi$ as $R(x)+2=2\alpha
\delta(\vec{x}-\vec{x}')$. The resulting Euclidean metric suffered from a
conical singularity at $\vec{x}=\vec{x}'$. A possible geometry modeled locally
in polar coordinates $(r,\varphi)$ by $ds^2=dr^2+r^2d\varphi^2,\varphi \cong
\varphi+2\pi-\alpha$. In this letter we showed that there exists another family
of "exact" geometries for arbitrary values of the $\alpha$. A pair of exact
solutions are found for the case of $\alpha=0$. One represents the static patch
of the AdS and the other one is the non static patch of the AdS metric. These
solutions were used to construct the Green function for the inhomogeneous model
with $\alpha\neq 0$. We address a type of the phase transition between
different patches of the AdS in theory because of the discontinuity in the
first derivative of the metric at $x=x'$. We extended the study to the exact
space of metrics satisfying the constraint
$R(x)+2=2\sum_{i=1}^{k}\alpha_i\delta^{(2)}(x-x'_i)$ as a modulo
diffeomorphisms for an arbitrary set of the deficit parameters
$(\alpha_1,\alpha_2,..,\alpha_k)$. The space is the moduli space of Riemann
surfaces of genus $g$ with $k$ conical singularities located at $x'_k$ denoted
by $\mathcal{M}_{g,k}$.
