What are explicit forms of $F(x,y)$ and $G(x,y)$ if $F(x,y)+iG(x,y)=\arcsin(x+iy)$.

Question

This relates to an earlier question and it answers in MSE:

$\frac{x^2}{a^2+ \lambda}+ \frac{y^2}{b^2+\lambda}=1$ family with $-a^2< \lambda < -b^2$ orthogonal to family with $\lambda > -b^2>-a^2$

In this regard, I have been trying to find explicit forms of $F(x,y)$ and $G(x,y)$ such that $$F(x,y)+iG(x,y)= \arcsin(x+iy)$$, they ought to be orthogonal but I wonder if they confocal pairs of elipse and hyperbola or they are deceptively so. Any comment or help is most welcome. I plot here $F(x,y)=\pm 1, G(x,y)=\pm 1$ for a visual aid.