Let $\{X_n\}$ be a Markov chain with state space $\mathbb{X}$ and stochastic matrix $P$. If $\{X_n\}$ is irreducible and $\mathbb{X}$ is finite, then it has a unique invariant.
My question is: can we proceed to finding the invariant $\pi$ using the conditions of detailed balance:
$$ \pi(x)p(x,y) = \pi(y)p(y,x), \quad \forall (x,y)\in \mathbb{X}^2 \quad \quad (1) $$
or are there any other assumptions that have to be made before?
I know
$$ \pi = \pi P, \, \sum_{x \in \mathbb{X}} \pi(x)=1 $$
is the standard way to go, but equations $(1)$ seem a lot easier to solve, so I'd like to know if it's proper to use them in this case.
a positive recurrent chain with a single communicating class will always have an invariant distribution that satisfies the global balance equations. It is of course a lot easier to calculate using the simpler detailed balance equations -- but such a chain will satisfy those iff it is time reversible. This is in fact the standard definition of time reversibility.