= Solution
The defining function $x^3+xy-1$ is independent of $z$, so a <normal vector> is $(3x^2+y,x,0)$. It never vanishes here because $x>0$. Every normal is horizontal, whereas $\mathbf G=(0,0,x)$ is vertical and tangent to the surface. Their <dot product> is zero at every point, giving
$$
\boxed{\int_{S_2}\mathbf G\cdot d\mathbf S=0.}
$$
This is independent of orientation. Although the surface extends without a bound on $x$, the integrand vanishes identically, so no convergence problem arises for this flux.
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