Deformed dessins d'enfants of almost Belyi maps

Almost Belyi maps are algebraic maps to $P^1\{0,1,\infty\}$ with exactly one (simple) branching point. As I explain from sratch, almost Belyi maps form 1-dimensional families. Their degenerations to Belyi maps are defined by image of braid monodromy as a Belyi map from the 1-dimensional base curve. I describe the geometric analogue of dessins d'enfants corresponding to almost Belyi maps. Special almost Belyi maps give isomonodromic Fuchsian differential equations corresponding to algebraic solutions of the Painleve VI equation.