  In this paper, we study positive integer solutions to a generalized form of
the Markov equation, given as $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 +
3k)xyz$. This equation extends the classical Markov equation $x^2 + y^2 + z^2 =
3xyz$. We generalize the concept of Cohn triples for the classical Markov
equation to the generalized Markov equations. Using this, we provide a
generalization of the uniqueness theorem of Markov numbers that are prime
powers.
