Stationary distribution of convex combination of Markov chains

Question

Let $P$ be a stochastic matrix (of an irreducible Markov Chain) with stationary distribution $\pi^T$ (i.e. $\pi^T P = \pi^T$) and let further $E$ be the matrix of all $1$'s.

Given an $\alpha \in [0,1]$, is it possible to find an expression for the stationary distribution of $$\alpha P + \frac{(1-\alpha)}{n}E,$$ depending on $\pi$ and $\frac{1}{n}\mathbb{1}$, where $\mathbb{1}$ is the vector of all $1$'s?

More generally; given two transition matrices of irreducible Markov Chains $P_1$ and $P_2$ with stationary distributions $\pi_1^T$ and $\pi_2^T$, respectively. Can one find a general formula to calculate the stationary distribution of $$\alpha P_1 + (1-\alpha)P_2 \quad,$$ for $\alpha \in [0,1]$?