I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres).
Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. Then
($i$) $n=2m$ for some $m \in \mathbb{Z}^{+}$ and
($ii$) there exist a holomorphic map $\psi: \mathbb{C}\mathbb{P}^1 \longrightarrow (I_m,g')$ which is horizontal with respect to the natural projection $\pi: (I_m,g')\longrightarrow (S^{2m},g_{S^{2m}}) $ and the diagram
$$\begin{array}{cc} & I_m & \\ ~~~\psi\nearrow && \searrow \pi~~~~\\\mathbb{C}\mathbb{P}^1 & \underset{\large \phi}{\longrightarrow} & \overset{\phantom{-}}{S^{2m}} \end{array}$$
commutes. Here $I_m$ is isotropic m-space.
And I saw the statement that this theorem tells us that the differential equations for $\phi$ can be transformed into first order equation for $\psi$. I don't understand this statement.
Could someone explain it to me?
Could someone help me to find or calculate how this first order equation looks like in coordinates?