Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections

Abstract. Isomonodromic deformations of rank 2 logarithmic connections with singular points 0, 1, t and ∞ over the Riemann sphere are parametrized by the solutions q(t) of Painleve ́ VI equa-tion. Some discrete group of symetries of PV I equation naturally arise from the birational geometry of logarithmic connections. An extra symmetry was found by Okamoto in [15] by direct compu-tations. Here, we present a geometric interpretation of this sym-metry. After lifting conveniently the connection over the elliptic curve Et: {y 2 = x(x − 1)(x − t)}, the variation of the underly-ing vector bundle (along isomonodromic deformation) provides a new solution q̃(t) of PV I equation, namely the Okamoto symetric of q(t). In particular, isomonodromic deformations of the Lamé connection over Et arise as a particular case of our construction and we recover in a natural way recent results of S. Kawai and Levin-Olshanetsky. All proofs will appear in a forthcoming pape