In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of nonlinear descent. Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics. The result is reminiscent of the finiteness of class numbers for linear arithmetic group actions on homogeneous varieties, and this Diophantine perspective guides us to obtain new extensions of classical results on hyperbolic surfaces along the way.