  We study generic conformally flat (analytic-)hypersurfaces in the Euclidean
$4$-space $\mathbb{R}^4$. Such a local-hypersurface is obtained as an evolution
of surfaces issuing from a certain surface in $\mathbb{R}^4$, and then, in
consequence, the original surface is a (principal-)curvature surface of the
hypersurface. The Poincar\'{e} metric ${\check g}_H$ of the upper half plane
leads to a $6$-dimensional set of rational Riemannian metrics $g_0$ of
$\mathbb{R}^2$: on a simply connected open set in the regular domain of $g_0$,
a curvature surface $f^0$ with the metric $g_0$ is determined, which we denote
by $(f^0,g_0)$. In this paper, we choose a suitable metric $g_0$ of
$\mathbb{R}^2$ determined by ${\check g}_H$ to get nice curvature surfaces (but
it also has degenerate and divergent points in $\mathbb{R}^2$), and clarify the
structure of the curvature surfaces $(f^0,g_0)$: the curvature surfaces
$(f^0,g_0)$ extend analytically to what kind of set in $\mathbb{R}^2$ beyond
the regular set of $g_0$, and then the extended surface $(f^0,g_0)$ is defined
on a certain open set of $\mathbb{R}^2$ and bounded in $\mathbb{R}^4$; for the
extended surface $(f^0,g_0)$, we explicitly catch the set of degenerate points
and the limits in $\mathbb{R}^4$ of both ends of every principal curvature
line, and then the two limits of every line for one principal curvature are
parallel small circles in a standard $2$-sphere $\mathbb{S}^2$. Then, every
principal curvature line in the extended surface $(f^0,g_0)$ is expressed by a
frame field of $\mathbb{R}^4$ induced on the surface from a hypersurface and it
lies on a standard $2$-sphere $\mathbb{S}^2$ with line-dependent radius. We
also provide a general method of constructing an approximation of such frame
fields, and obtain the entire pictures of those lines including degenerate
points of $(f^0,g_0)$.
