The equations are symmetric in x and y, so I will try to use that to my advantage.

Start by squaring each side:

x^2 + 2xy + y^2 = 529

x^2 + 12*y + y^2 + 12*x + 2*sqrt[(xy)^2 + 12*y^3 + 12*x^3+ 144xy] = 1089

Rearrange and group into terms of (x^2+y^2), (x+y), and (xy):

(x^2+y^2) = 529-2xy

(x^2+y^2) + 12(x+y) + 2*sqrt[(xy)^2 + 12(x+y)(x^2+y^2-xy) + 144xy] = 1089

Now substitute (x^2+y^2) = 529-2xy and x+y=23 into the last equation to get an equation in terms of xy:

529-2xy + 12*23 + 2*sqrt[(xy)^2 + 12*23*(529-3xy) + 144*xy] = 1089

Isolate the sqrt expression and simplify:

2*sqrt[(xy)^2 + 146004 - 828xy + 144*xy] = 1089 - 529 + 2xy - 276

sqrt[(xy)^2 - 684*xy + 146004] = 142 + xy

Square each side and simplify:

(xy)^2 - 684*xy + 146004 = (xy)^2 + 284xy + 20164

968xy = 125840

xy = 130

Then the system xy=130, x+y=23 yields two solutions **(x,y) = (10,13) and (13,10)**.