On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. Moduli spaces of these objects are obtained with fixed generic polar parts at each singularity, which amounts to fixing a coadjoint orbit of the group GL r (C[z]/z n ). We prove that they carry complete hyper-Kähler metrics.

Definition 2.5. The moduli space M * (a) is the set of isomorphism classes of pairs (V, ∇) where V is a trivial rank n holomorphic vector bundle over P 1 and ∇ is a meromorphic connection on V which is formally equivalent to d − i A 0 at a i for each i and has no other poles.Following [40], we also define slightly larger moduli spaces:Definition 2.6. The extended moduli space M * (a) is the set of isomorphism classes of triples (V, ∇, g) consisting of a generic connection ∇ (with poles on D) on a trivial holomorphic vector bundle V over P 1 with compatible framings g = ( 1 g 0 , . . . , m g 0 ) such that (V, ∇, g) has irregular type i A 0 at each a i .The term 'extended moduli space' is taken from the paper [37] of L.Jeffrey, since these spaces play a similar role (but are not the same).Remark 2.1. For use in later sections we also define spaces M(a) and M(a) simply by replacing the word 'trivial' by 'degree zero' in Definitions 2.5 and 2.6 respectively.

Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL 2 (C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin-Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL 2 (C).The results of this paper also yield a simple proof of a recent theorem of InabaIwasaki-Saito on the action of Okamoto's affine D 4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations. IntroductionKlein's quartic curveis of genus three and has the maximum possible number 84(g − 1) = 168 of holomorphic automorphisms. Klein found these automorphisms explicitly (in terms of 3 × 3 matrices). They constitute Klein's simple group K ⊂ PGL 3 (C) which is isomorphic to PSL 2 (7). Lifting to GL 3 (C) there is a two-fold covering group K ⊂ GL 3 (C) of order 336 which is a complex reflection group-there are complex reflections (1) r 1 , r 2 , r 3 ∈ GL 3 (C) which generate K. (Recall a pseudo-reflection is an automorphism of the form "one plus rank one", a complex reflection is a pseudo-reflection of finite order and a complex reflection group is a finite group generated by complex reflections. Here, each generator r i has order two-as for real reflections.) Using the general tools to be described in this paper we will construct, starting from the Klein complex reflection group K, another algebraic curve with affine equation (2) F (t, y) = 0given by a polynomial F with integer coefficients. This curve will be a seven-fold cover of the t-line branched only at 0, 1, ∞ and such that the function y(t), defined implicitly by (2), solves the Painlevé VI differential equation. One upshot of this will be to construct an explicit rank three Fuchsian system of linear differential equations with four singularities (at 0, t, 1, ∞, for some t) on P 1 , and with 1 2 PHILIP BOALCH monodromy group equal to K in its natural representation (so the monodromy around each of the finite singularities 0, t, 1 is a generating reflection).In general the construction of linear differential equations with finite monodromy group is reasonably straightforward provided one works with rigid representations of the monodromy groups. In our situation the representation is minimally non-rigid; it lives in a complex two-dimensional moduli space, and this is the basic reason the (second order) Painlevé VI equati...

The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F 4 Weyl group action and many properties of the solutions will be given.There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known P VI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev.Of the remaining 28 solutions we will find 20 explicitly using the method of [5] (via Jimbo's asymptotic formula). Amongst those constructed there is one solution that is "generic" in that its parameters lie on none of the affine F 4 hyperplanes, one that is equivalent to the Dubrovin-Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.