$x \in R^n$, $P \in R^{n \times n}$, $P$ is a known stochastic matrix (each row sums to 1), $b \in R^{n}$ is a known non-zero vector. $e = [1, \dots, 1]^T \in R^n$, we have the linear system:
$x = P^Tx + b$
$e^Tx=0$
Does this linear system have a unique solution?
Added: Assuming we already know one solution for this linear system, is this a unique solution?