Minimal surfaces in AdS space and Integrable systems

We consider the Pohlmeyer reduction for spacelike minimal area worldsheets in AdS${}_5$. The Lax pair for the reduced theory is found, and written entirely in terms of the $A_3=D_3$ root system, generalizing the $B_2$ affine Toda system which appears for the AdS${}_4$ string. For the $B_2$ affine Toda system, we show that the area of the worlsheet is obtainable from the moduli space K"ahler potential of a related Hitchin system. We also explore the Saveliev-Leznov construction for solutions of the $B_2$ affine Toda system, and recover the rotationally symmetric solution associated to Painleve transcendent.