solve: Uxy = -Ux using indicated transformation v=x and z=x+y

I think you must be missing a lot of information. For one thing, you can solve this PDE using direct integration...

But if you must transform, use the fact that, for any function, F(x,y), $\frac{\partial F}{\partial x}= \frac{\partial F}{\partial v}\frac{\partial v}{\partial x}+ \frac{\partial F}{\partial z}\frac{\partial z}{\partial x}= \frac{\partial F}{\partial v}+ \frac{\partial F}{\partial z}$ and $\frac{\partial F}{\partial y}= \frac{\partial F}{\partial v}\frac{\partial v}{\partial y}+ \frac{\partial F}{\partial z}\frac{\partial z}{\partial y}= \frac{\partial F}{\partial z}$
So, letting $F= U$, $U_x= U_v+ U_z$ and, letting $F= U_v+ U_z$, $U_{xy}= U_{vz}+ U_{zz}$.
The equation $U_{xy}= -U_x$ becomes $U_{vz}+ U_{zz}= -U_v- U_z$.