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\begin{document}
    \pagenumbering{Roman}
Plugging in
\newline
\begin{align}
\begin{split}
x_t&=\frac{\sigma-\delta_x}{\sigma}(\omega^{x,ext}_t(1+\alpha_x))x_t +(\omega^{x,sta}_t-\alp\omega^{x,ext})x_{t-1}
&+\frac{1-\delta_\pi}{\sigma}(\omega^{\pi, ext}_t(1+\alpha_\pi))\pi_t 
&+(\omega^{\pi,sta}_t - \alpha_\pi\omega^{\pi,ext}_t)\pi_{t-1}-\frac{k}{\sigma}+u_t
\end{split}
\end{align}
\newline
After rearranging
\begin{align}
(1-\beta\omega^{\pi,ext}_t(1+\alpha_\pi))\pi_t- \gamma x_t &= \beta(\omega^{\pi,sta}_t -\alpha_\pi\omega^{\pi,ext})\pi_{t-1} +v_t
\end{align}
\begin{align}
\begin{split}
x_t-(\frac{\sigma-\delta_x}{\sigma})(\omega^{x,ext}_t(1+\alpha_x))x_t
-\frac{1-\delta_\pi}{\sigma}(\omega^{\pi,ext}_t(1+\alpha_\pi))\pi_t &=\frac{\sigma-\delta_x}{\sigma}(\omega^{x,sta}_t-\alpha_x\omega^{x,ext}_t)x_{t-1}
&+(\frac{1-\delta_\pi}{\sigma})(\omega^{\pi,sta}_t-\alpha\pi^{\pi,sta})\pi_{t-1}-\frac{k}{\sigma} +v_t
\end{split}
\end{align}
\begin{align}
\begin{split}
(1-(\frac{\sigma-\delta_x}{\sigma})(\omega^{x,sta}_t-\alpha_x\omega^{x,ext})x_t -\frac{1-\delta_\pi}{\sigma}(\omega^{x,ext}_t(1+\alpha_\pi)\pi_t &=\frac{\sigma-\delta_x}{\sigma}(\omega^{x,sta}_t-\alpha_x\omega^{x,ext}_t)x_{t-1}
&+(\frac{1-\delta_\pi}{\sigma})(\omega^{\pi,sta}_t-\alpha\pi^{\pi,sta})\pi_{t-1}-\frac{k}{\sigma} +v_t
\end{split}
\end{align}
\begin{align}
\begin{split}
(\sigma-(\sigma-\delta_x)(\omega^{x,sta}_t(1+\alpha_x))x_t-(1-\delta_\pi)(\omega^{\pi,ext}_t+(1+\alpha_\pi))\pi_t 
&=\sigma-\sigma_x(\omega^{x,sta}_t-\alpha_x\omega^{x,ext}_t)x_{t-1} 
&+(1-\delta_\pi)(\omega^{\pi,sta}_t-\alpha_\pi\omega^{\pi,ext})\pi_{t-1}- k + u_t\sigma
\end{split}
\end{align}
\newline
and the \textit{time independent} matrices
\begin{equation}    \label{eq:Matrix}
 \bm{A}_t =
 \begin{bmatrix} 
 1 - \beta\omega^{\pi, ext}_t(1 + \alpha_\pi) &  -\gamma \\
  -(1-\delta\pi)(\omega^{\pi,ext}(1+\alpha_\pi))&\sigma-(\sigma-\delta_x)(\omega^{x,ext}_t(1+\alpha_x))\\
 \end{bmatrix}
$ 
$ \bm{B}_t =
  \begin{bmatrix}
  \beta(\omega^{\pi, sta}_t -\alpha_\pi \omega^{\pi, ext}_t) & 0 \\
 (\omega^{\pi, sta}_t  - \alpha_\pi \omega^{\pi, ext}_t) & \sigma(\omega^{\pi, sta}_t - \alpha_x \omega^{\pi, ext}_t) \\
$$\end{bmatrix}
$ \end{equation}
\newline
\newline
\newline
Multiplying with $A^{-1}$ we derive the solution:
\newline
\begin{equation}   \label{eq:SolutionBRD}
   \begin{bmatrix}  
       \pi_t\\
         x_t \\
         \end{bmatrix}
         = \bm{A}_t^{-1} \bm{B}_t  \begin{bmatrix} 
   \pi_{t-1}\\
         x_{t-1} \\      
   \end{bmatrix}
   + \bm{A}_t^{-1} 
    \begin{bmatrix} 
      \upsilon_t\\
         \sigma u_t -k_t
       \end{bmatrix}
\end{equation}
\newline
\end{document}