  It is known that complex constant mean curvature ({\sc CMC} for short)
immersions in $\mathbb C^3$ are natural complexifications of {\sc
CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it
real form surfaces} of a complex {\sc CMC}-immersion, which are defined from
real forms of the twisted $\mathfrak{sl}(2, \mathbb C)$ loop algebra $\Lambda
\mathfrak{sl}(2, \mathbb C)_\sigma$, and classify all such surfaces according
to the classification of real forms of $\Lambda \mathfrak{sl}(2, \mathbb
C)_\sigma$. There are seven classes of surfaces, which are called {\it
integrable surfaces}, and all integrable surfaces will be characterized by the
(Lorentz) harmonicities of their Gau{\ss} maps into the symmetric spaces $S^2$,
$H^2$, $S^{1,1}$ or the 4-symmetric space $SL(2, \mathbb C)/U(1)$. We also give
a unification to all integrable surfaces via the generalized Weierstra{\ss}
type representation.
