Three, given the system of equations $\left\{\begin{array}{l}x-y=2, \\ m x+y=6 .\end{array}\right.$ If the system has non-negative integer solutions, find the value of the positive integer $m$, and solve the system of equations.
Three, adding the two equations, we get $(m+1) x=8$.
Since the system of equations has non-negative integer solutions, and $m$ is a positive integer, the possible values of $m$ are $1,3,7$.
When $m=1$, we have $x=4$, solving for $y=2$.
When $m=3$, we have $x=2$, solving for $y=0$. When $m=7$, we have $x=1$. Solving for $y=-1$ (discard). Therefore, the solutions to the system of equations that satisfy the conditions are