  The classical result of describing harmonic maps from surfaces into symmetric
spaces of reductive Lie groups states that the Maurer-Cartan form with an
additional parameter, the so-called loop parameter, is integrable for all
values of the loop parameter. As a matter of fact, the same result holds for
$k$-symmetric spaces over reductive Lie groups. In this survey we will show
that to each of the five different types of real forms for a loop group of
$A_2^{(2)}$ there exists a surface class, for which some frame is integrable
for all values of the loop parameter if and only if it belongs to one of the
surface classes, that is, minimal Lagrangian surfaces in $\mathbb {CP}^2$,
minimal Lagrangian surfaces in $\mathbb {CH}^2$, timelike minimal Lagrangian
surfaces in $\mathbb {CH}^2_1$, proper definite affine spheres in $\mathbb R^3$
and proper indefinite affine spheres in $\mathbb R^3$, respectively.
