Elliptic Linear Problem for the Calogero—Inozemtsev Model and Painlevé VI Equation

We introduce a 3N × 3N Lax pair with spectral parameter for the Calogero—Inozemtsev model. The case of one degree freedom appears to have 2 × 2 Lax representation. We derive it from the elliptic Gaudin model via some reduction procedure and prove algebraic integrability. This Lax pair provides an elliptic linear problem for the Painlevé VI equation in elliptic form.