  In this paper, we deal with two classes of Diophantine equations,
$x^2+y^2+z^2+k_1yz+k_2zx+k_3xy=(3+k_1+k_2+k_3)xyz$ and
$x^2+y^4+z^4+ky^2z^2+2xz^2+2xy^2=(7+k)xy^2z^2$, 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.
