Generalization of Markov Diophantine equation via generalized cluster algebra

In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_{1}yz+k_{2}zx+k_{3}xy=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+k{y}^2{z}^2+2x{z}^2+2x{y}^2=(7+k)x{y}^2{z}^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.