Simply-laced isomonodromy systems

A new class of isomonodromy equations will be introduced and shown to admit Kac?CMoody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac?CMoody Weyl groups and root systems occur ??in nature??. A key point is that one may go beyond the class of affine Kac?CMoody root systems. As examples, by considering certain hyperbolic Kac?CMoody Dynkin diagrams, we find there is a sequence of higher order Painlevé systems lying over each of the classical Painlevé equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.