  A starting point of this paper is a classification of quadratic polynomial
transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian
systems associated to the Painleve VI equation. Up to birational automorphisms
of the monodromy manifold, we find three transformations. Two of them are
identified as the action of known quadratic or quartic transformations of the
Painleve VI equation. The third transformation of the monodromy manifold gives
a new transformation of degree 3 of Picard's solutions of Painleve VI.
