  We consider the Pohlmeyer reduced sigma model describing AdS$_5$ minimal
surfaces. We show that, similar to the affine Toda models, there exists a
conformal extension to this model which admits a Lax formulation. The Lax
connection is shown to be valued in a ${\mathbb Z}_4$-invariant subalgebra of
the affine Lie algebra $\widehat{su(4)}$. Using this, we perform a modified
version of a Laznov-Savelev analysis, which allows us to write formal
expressions for the general solutions for the Pohlmeyer reduced AdS$_5$ theory.
This analysis relies on the a certain decomposition for the exponentiated
algebra elements.
