  It has been known for some time that there exist $5$ essentially different
real forms of the complex affine Kac-Moody algebra of type $A_2^{(2)}$ and that
one can associate $4$ of these real forms with certain classes of "integrable
surfaces", such as minimal Lagrangian surfaces in $\mathbb {CP}^2$ and $\mathbb
{CH}^2$, as well as definite and indefinite affine spheres in $\mathbb R^3$.
  In this paper we consider the class of timelike minimal Lagrangian surfaces
in the indefinite complex hyperbolic two-space $\mathbb{CH}^2_1$. We show that
this class of surfaces corresponds to the fifth real form.
  Moreover, for each timelike Lagrangian surface in $\mathbb {CH}^2_1$ we
define natural Gauss maps into certain homogeneous spaces and prove a Ruh-Vilms
type theorem, characterizing timelike minimal Lagrangian surfaces among all
timelike Lagrangian surfaces in terms of the harmonicity of these Gauss maps.
