Simply-laced isomonodromy systems

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