Parametric Equation solving over integers

Question

I have a question on my mind I am trying to solve. However I am stuck at a point. If you could help, I would be very pleased. $$\frac {x_2y_2z_2}{x_2y_2+y_2z_2+x_2z_2}=\frac {x_1y_1z_1}{x_1y_1+y_1z_1+x_1z_1}+6 $$ I have tried to solve this equation however I couldn't, by solving I mean finding $x_2,y_2,z_2$ in terms of $x_1,y_1,z_1$, and all of the terms must be integers. For instance for observation I tried doing it when $x_1=y_1=z_1$ and on solution became $x_2=x_1+18$ and similar for $y_1, z_1$. However I seemed to come to the conclusion that all three must be different: $x_1\neq x_2\neq x_3$, we can find solutions such as $x_2= y_1z_1+x_1+7$ however all of the terms $x_1,x_2, y_1,y_2, z_1,z_2$ must be positive integers. Thanks in advance!