  Several kinds of differential relations for polynomial components of almost
Belyi maps are presented. Saito's theory of free divisors give particularly
interesting (yet conjectural) logarithmic action of vector fields. The
differential relations implied by Kitaev's construction of algebraic Painleve
VI solutions through pull-back transformations are used to compute almost Belyi
maps for the pull-backs giving all genus 0 and 1 Painleve VI solutions in the
Lisovyy-Tykhyy classification.
