  We classify all functions satisfying non-trivial families of PVI equations.
It turns out that, up to an Okamoto equivalence, there are exactly four
families parameterized by affine planes or lines. Each affine space is
generated by points of "geometric origin", associated either to deformations of
elliptic surfaces with four singular fibers, or to deformations of
three-sheeted covers of the projective line with branching locus consisting of
four points.
