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Abstract:For the projective line over complex numbers with 4 punctures 0, 1, \infity, \lambda, where lambda is a parameter running in P^-{0, 1, \infty} we construct infinitely many rank-2 local systems with the prescribed local monodromy around those 4 punctures, which come from geometry origin. Consequently, they all are algebraic solutions of the Painleve VI equation. The method is totally d...