My method is to use characteristics.
$$dx/(y+z)=dy/(z+x)=dz/(x+y).$$ Solve this??
$d(x+y)/(x+y+2z)=dz/(x+y)$ which implies $$(x+y-2z)^2(x+y+z)=C_1.$$
And by symmetry, another independent first integral is $$(y+z-2x)^2(x+y+z)=C_2$$ and thus the solution of the PDE is $$u=\phi((x+y-2z)^2(x+y+z),(y+z-2x)^2(x+y+z)).$$Is this right?