Probability density function (PDF) of a two-dimensional linear stochastic differential equation (SDE) with additive noise

Question

I am considering the same two-dimensional linear SDE as in a previous question:

\begin{cases} dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1, \\ dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \bigr)\,dt + \sigma \,dW_2, \end{cases}

where $W_1$ and $W_2$ are one-dimensional Wiener processes, the $A_{ij}$ are real, and so is $\sigma$.

Is a there a way to obtain an analytical expression for the (unconditional) PDF of $X_1$? Thank you for your help.