Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry

A Lamé connection is a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection $(E,\nabla)$ over an elliptic curve $X\colon \{y^2=x(x-1)(x-t)\}$, $t\neq 0,1$, having a single pole at infinity. When this connection is irreducible, we show that it is invariant under the standard involution and can be pushed down to a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection on $\mathbb P^1$ with poles at $0$, $1$, $t$ and $\infty$. Therefore the isomonodromic deformation $(E_t,\nabla_t)$ of an irreducible Lamé connection, when the elliptic curve $X_t$ varies in the Legendre family, is parametrized by a solution $q(t)$ of the Painlevé VI differential equation $\mathrm{P}_{\mathrm{VI}}$. The variation of the underlying vector bundle $E_t$ along the deformation is computed in terms of the Tu moduli map: it is given by another solution $\tilde q(t)$ of $\mathrm{P}_{\mathrm{VI}}$, which is related to $q(t)$ by the Okamoto symmetry $s_2 s_1 s_2$ (Noumi–Yamada notation). Motivated by the Riemann–Hilbert problem for the classical Lamé equation, we raise the question whether the Painlevé transcendents do have poles. Some of these results were announced in [6].