Is there some known parametric form for modeling a distribution of transition probability matrices given the leading left eigenvector?

Question

Let $X$ be a first-order Markov chain and $A = [A_{ij}]_{i,j=1}^K$ be its transition probability matrix, i.e. $A_{ij} = Pr(X_{t} = j \, | \, X_{t−1} = i)$. Suppose further that $X$ is irreducible and aperiodic, so a stationary distribution of $X$ exists and is unique. Then that stationary distribution, $\pi$, is given by the leading left-eigenvector of $A$.

Now suppose we know $\pi$ but not $A$. Is there some known parametric family that can be used to model the probability distribution $Pr( A \, | \, \pi)$?