Moduli spaces of local systems on surfaces are widely studied in geometry. Focusing on the special linear rank two case, after motivating our Diophantine study we use mapping class group dynamics and differential geometric tools to establish a structure theorem for the integral points on the moduli spaces, generalizing work of Markoff (1880). We also give an effective analysis of integral points for nondegenerate algebraic curves on these spaces. Along the way, we present other related results connecting the geometry and arithmetic of the moduli spaces to elementary observations on surfaces