Three. (50 points) Try to determine all positive integer solutions of the indeterminate equation
$$
x^{2}+y^{2}-5=3 x y
$$
Three, it is easy to see that $(1,4)$ is a set of positive integer solutions to the indeterminate equation (1).

Let $(a, b)$ be a set of positive integer solutions to the indeterminate equation (1). Then $(b, a)$ is also a set of positive integer solutions.
Without loss of generality, assume $a<b$.
It is easy to know that $(a, 3a-b)$ and $(3b-a, b)$ are also integer solutions to the indeterminate equation (1).

In general, let $L_{1}=1, L_{2}=3, L_{n}=L_{n-1}+L_{n-2}$. Then it can be proven that
$$
\begin{array}{l}
L_{2 n-1}^{2}+L_{2 n+1}^{2}-5=3 L_{2 n-1} L_{2 n+1} . \\
\text { Thus, }\left(L_{2 n-1}, L_{2 n+1}\right)(n=1,2, \cdots) \text { are all positive }
\end{array}
$$

integer solutions to the indeterminate equation (1), and by mathematical induction, it can be proven that $\left(L_{2 n-1}, L_{2 n+1}\right)$ or $\left(L_{2 n+1}, L_{2 n-1}\right)$ are all the positive integer solutions.