# Finding stationary distribution of random process

Suppose we are given $$x_t, \bar{x_t}, t\in \mathbb{Z_+}$$ independent 2-states $$\{0, 1\}$$ Markov chains with positive transition probabilities. Initial states are $$x_0 = 0; \bar{x}_0 = 1$$. For which positive real numbers $$a, b$$ random process $$\eta_t = ax_t + b\bar{x_t}$$ is stationary?

I know that stationary distribution for one particular Markov chain would be $$\pi = (0.5, 0.5)$$. How to proceed from here? Or instead i need to check that distribution of $$\eta _t$$ is translation invariant?