Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 + 4 z^2 = 1$$
Here, suppose $g = \frac{dx^2 + dy^2 + dz^2}{x^2}$ and I'd like $Y^2$ minimal to be compact in the euclidean sense, i.e. lying in $\{(x, y,z) \; | \; y^2 + z^2 \leq R\}$ for some $R$. By "Explicit parameterization", I mean a computable chart so that one could integrate over $Y^2$, the asymmetric minimal surface.
