Too partial

Is there any solution of $f_{xx}-f_{yy}+f_{zz}-2f_{xz}=(y+z)(x+z)$ satisfying $f_y+f_z-f_x=0$ on all the points of the plane $x+y=0$ and $f=0$ on all the points of the plane $x=0$ ?

Bonus : find such set of solutions.