Cubic and Quartic Transformations of the Sixth Painleve Equation in Terms of Riemann-Hilbert Correspondence

The starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 x 2 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 Picards solutions of Painleve VI.