Weird problem with a Lagrangian

[ShayanJ]

I'm trying to follow the calculations in this paper. But I have a weird problem in section 2.
To calculate the entanglement entropy using the Ryu-Takayanagi prescription, you have to extremize the area of a surface. So you have to use Euler-Lagrange equations for some kind of an action. The authors don't do all the calculations and just point out that the result is eq. 13.
But when I do the calculations I get something else. Actually eq. 13 doesn't even satisfy the Euler-Lagrange equation of the action but my solution does. So it makes me think that I may have done something wrong in getting the equations of motion but I'm pretty sure about my calculations, I've done and checked them more than a dozen times!
So I just have no idea what is wrong here. My calculations are attached.(Sorry if I don't write them here, but I made this pdf for my advisor so I just attach it here!)
Thanks

P.S.
I also think that in eq.10, the power of $\alpha$ should be 2 instead of d. But its irrelevant here because $H(U)$ is just assumed to be some known function.

 

[ShayanJ]

Another weird thing that I just realized. If you put $U_x=\sqrt{-\frac 1 \beta}$ in the action, you get zero!

EDIT: Or maybe there is nothing wrong with it. The on-shell value of the action can be anything and now its zero.