Consider two RVs $X_1$, $X_2$, where the density of $X_1$ is $p_{X_1}(\cdot)$ while $X_2 = X_1-x_0$ for some costant $x_0$, i.e. $X_2$ is a simple translation of $X_1$. I want to find, if it possible, the joint density $p_{X_1, X_2}(\cdot, \cdot)$ of $X_1$, $X_2$.
In order to do that, I start from the cumulative density $P_{X_1, X_2}(\cdot, \cdot)$:
\begin{equation}
\begin{aligned}P_{X_1, X_2}(x_1, x_2) &\triangleq \mathbb{P}(X_1 \leq x_1, X_2 \leq x_2)=\mathbb{P}(X_1 \leq x_1, X_1-x_0 \leq x_2) \\
&=\mathbb{P}(X_1 \leq x_1, X_1\leq x_2-x_0)=\mathbb{P}(X_1 \leq \text{min}(x_1, x_2-x_0))\\
&=\int_{-\infty}^{\text{min}(x_1, x_2-x_0)} p_{X_1}(\xi_1)\text{ d}\xi_1
\end{aligned}\end{equation}
on the other hand
\begin{equation}P_{X_1, X_2}(x_1, x_2) = \int_{-\infty}^{x_1}\int_{-\infty}^{x_2} p_{X_1, X_2} (\xi_1,\xi_2) \text{ d}\xi_1\text{ d}\xi_2\end{equation}
so
\begin{equation}p_{X_1, X_2} (x_1,x_2)=\frac{\partial^2}{\partial x_1\partial x_2} P_{X_1, X_2}(x_1, x_2)=\frac{\partial^2}{\partial x_1\partial x_2}\int_{-\infty}^{\text{min}(x_1, x_2-x_0)} p_{X_1}(\xi_1)\text{ d}\xi_1\end{equation}
but I stuck here. Maybe there is a more simple approach. I have the suspicion that the solution is something like
\begin{equation}p_{X_1, X_2} (x_1,x_2)=p_{X_1}(x_1)\, \delta_{X_1-x_0}(x_2) \end{equation}
where $\delta_k(\cdot)$ is the Dirac delta concentrated in some point $k$
The joint density does not exist since the joint distribution is supported by a the line $y=x-x_0$ in $\mathbb R^{2}$.
[Any line in $\mathbb R^{2}$ has two dimensional Lebesgue measure $0$].