The integrable structure of \(\varOmega \) -surfaces is induced from that of the enveloping isothermic sphere congruences. To better understand the integrability of isothermic sphere congruences, we make comparison between isothermic surfaces and isothermic sphere congruences. Recall that a point \(x \in \mathbb {R}^3\) can be lifted into \(X \in M_0 \subset L^4 \subset \mathbb {R}^{4,1}\) via \(\displaystyle \begin{aligned} X = \left(\tfrac{1}{2}(1 + |x|^2), x^t, \tfrac{1}{2}(1 - |x|^2)\right)^t, \end{aligned}\) where \(M_0\) is determined by the space form vector \(\mathfrak {q}_0 = (1, 0, 0, 0, -1)^t\) . Choosing \(\mathfrak {o} = \tfrac {1}{2} (1, 0, 0, 0, 1)^t\) so that \(\langle \mathfrak {o}, \mathfrak {q_0} \rangle = -1\) , and viewing x as a vector in \(\mathbb {R}^{4,1}\) via \((0, x^t, 0)^t\) , we see that \(\displaystyle \begin{aligned} X = x + \mathfrak{o} + \frac{1}{2} (x \cdot x) \: \mathfrak{q}_0. \end{aligned}\) We have further seen that the isothermicity of \(L = \operatorname {span}\{X\}\) in \(P(L^4)\) is equivalent to the isothermicity of x in \(\mathbb {R}^3\) .