The talk emphases interrelations between integrable billiards, extremal polynomials, Riemann surfaces, potential theory, and isomonodromic deformations. We introduce and study dynamics of Chebyshev polynomials on several intervals. We introduce a notion of isoharmonic deformations and study their isomonodromic properties. We formulate a new class of constrained Schlesinger systems and provide explicit solutions to these systems. One line of motivation for this research goes back to works of Hitchin in mid 90's. The talk is based on joint results with Vasilisa Shramchenko, including work in progress.