I found the following the following differential equation in the context of a Game Theory problem. I was wondering if this is related to any known family of equations or whether there is any hint about properties it might have. I am looking for functions $f:[0,1]^2 \rightarrow [0,1]^2$ satisfying:

$x_1 \frac{\partial f_1}{\partial x_1} + x_1 \frac{\partial f_2}{\partial x_1} = \frac{\partial f_2}{\partial x_1}$

$x_2 \frac{\partial f_1}{\partial x_2} + x_2 \frac{\partial f_2}{\partial x_2} = \frac{\partial f_1}{\partial x_2}$