Stationary distributions of two-state chains

The transition probability matrices of Markov chains with state space $S=\{1,2\}$ are

$$P_1=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_2=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_3=\begin{pmatrix}\tfrac12&\tfrac12\\\tfrac23&\tfrac13\end{pmatrix},\quad P_4=\begin{pmatrix}\tfrac12&\tfrac12\\0&1\end{pmatrix}.$$

Which Markov chains have a stationary distribution given by $\left(\tfrac12,\tfrac12\right)$?