By "isolated", I mean that each state of this Markov Chain has 0 probability to move to another state, i.e. transition probability $p_{ij} = 0$ for $ i \ne j$. Thus, there isn't a unique stationary distribution.
But by definition, since for any stationary distribution $\pi$, we have
$ \pi_{i}p_{ij} = 0 = \pi_{j}p_{ji} $
seems that we can still call this Markov Chain time reversible.
Is the concept "time reversible" still make sense in this situation?
A bit background, I was asked to find a Markov Chain, with certain restrictions, that is NOT time reversible. But I found if the stationary distribution exist, the chain is always reversible. So I guess that my be chance is that a chain who doesn't have unique stationary distribution. Maybe in this situation we can't call the chain reversible.