Uniqueness theorem of generalized Markov numbers that are prime powers

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.