  The solutions of the (nonlinear) Painleve VI differential equation having
icosahedral linear monodromy group will be classified up to equivalence under
Okamoto's affine F4 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 PVI 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 author's
correction of) Jimbo's asymptotic formula. Amongst those constructed there is
one solution that is 'generic' in that its parameters lie on none of the affine
F4 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.
