  Let $(X_n \colon n\in\Z)$ be a two-sided recurrent Markov chain with fixed
initial state $X_0$ and let $\nu$ be a probability measure on its state space.
We give a necessary and sufficient criterion for the existence of a
non-randomized time $T$ such that $(X_{T+n} \colon n\in\Z)$ has the law of the
same Markov chain with initial distribution $\nu$. In the case when our
criterion is satisfied we give an explicit solution, which is also a stopping
time, and study its moment properties. We show that this solution minimizes the
expectation of $\psi(T)$ in the class of all non-negative solutions,
simultaneously for all non-negative concave functions $\psi$.
